Exponential and Logarithmic Functions · ⁨פונקציות מעריכיות ולוגריתמיות⁩

Learning Objective 2.1.A: Express arithmetic sequences found in mathematical and contextual scenarios as functions of the whole numbers.

• 2.1.A.1 A sequence is a function from the whole numbers to the real numbers. Consequently, the graph of a sequence consists of discrete points instead of a curve.
• 2.1.A.2 Successive terms in an arithmetic sequence have a common difference, or constant rate of change.
• 2.1.A.3 The general term of an arithmetic sequence with a common difference $d$ is denoted by $a_n$ and is given by $a_n = a_0 + dn$, where $a_0$ is the initial value, or by $a_n = a_k + d(n - k)$, where $a_k$ is the $k$th term of the sequence.

Learning Objective 2.1.B: Express geometric sequences found in mathematical and contextual scenarios as functions of the whole numbers.

• 2.1.B.1 Successive terms in a geometric sequence have a common ratio, or constant proportional change.
• 2.1.B.2 The general term of a geometric sequence with a common ratio $r$ is denoted by $g_n$ and is given by $g_n = g_0 r^n$, where $g_0$ is the initial value, or by $g_n = g_k r^{(n-k)}$, where $g_k$ is the $k$th term of the sequence.
• 2.1.B.3 Increasing arithmetic sequences increase equally with each step, whereas increasing geometric sequences increase by a larger amount with each successive step.

Learning Objective 2.2.A: Construct functions of the real numbers that are comparable to arithmetic and geometric sequences.

• 2.2.A.1 Linear functions of the form $f(x) = b + mx$ are similar to arithmetic sequences of the form $a_n = a_0 + dn$, as both can be expressed as an initial value ($b$ or $a_0$) plus repeated addition of a constant rate of change, the slope ($m$ or $d$).
• 2.2.A.2 Similar to arithmetic sequences of the form $a_n = a_k + d(n - k)$, which are based on a known difference, $d$, and a $k$th term, linear functions can be expressed in the form $f(x) = y_i + m(x - x_i)$ based on a known slope, $m$, and a point, $(x_i, y_i)$.
• 2.2.A.3 Exponential functions of the form $f(x) = ab^x$ are similar to geometric sequences of the form $g_n = g_0 r^n$, as both can be expressed as an initial value ($a$ or $g_0$) times repeated multiplication by a constant proportion ($b$ or $r$).
• 2.2.A.4 Similar to geometric sequences of the form $g_n = g_k r^{(n-k)}$, which are based on a known ratio, $r$, and a $k$th term, exponential functions can be expressed in the form $f(x) = y_i r^{(x - x_i)}$ based on a known ratio, $r$, and a point, $(x_i, y_i)$.
• 2.2.A.5 Sequences and their corresponding functions may have different domains.

Learning Objective 2.3.A: Identify key characteristics of exponential functions.

• 2.3.A.1 The general form of an exponential function is $f(x) = ab^x$, with the initial value $a$, where $a \neq 0$, and the base $b$, where $b > 0$, and $b \neq 1$. When $a > 0$ and $b > 1$, the exponential function is said to demonstrate exponential growth. When $a > 0$ and $0 < b < 1$, the exponential function is said to demonstrate exponential decay.
• 2.3.A.2 When the natural numbers are input values in an exponential function, the input value specifies the number of factors of the base to be applied to the function's initial value. The domain of an exponential function is all real numbers.
• 2.3.A.3 Because the output values of exponential functions in general form are proportional over equal-length input-value intervals, exponential functions are always increasing or always decreasing, and their graphs are always concave up or always concave down. Consequently, exponential functions do not have extrema except on a closed interval, and their graphs do not have points of inflection.
• 2.3.A.4 If the values of the additive transformation function $g(x) = f(x) + k$ of any function $f$ are proportional over equal-length input-value intervals, then $f$ is exponential.

Learning Objective 2.4.A: Rewrite exponential expressions in equivalent forms.

• 2.4.A.1 The product property for exponents states that $b^m b^n = b^{(m+n)}$. Graphically, this property implies that every horizontal translation of an exponential function, $f(x) = b^{(x+k)}$, is equivalent to a vertical dilation, $f(x) = b^{(x+k)} = b^x b^k = ab^x$, where $a = b^k$.
• 2.4.A.2 The power property for exponents states that $\left(b^m\right)^n = b^{(mn)}$. Graphically, this property implies that every horizontal dilation of an exponential function, $f(x) = b^{(cx)}$, is equivalent to a change of the base of an exponential function, $f(x) = \left(b^c\right)^x$, where $b^c$ is a constant and $c \neq 0$.
• 2.4.A.3 The negative exponent property states that $b^{-n} = \dfrac{1}{b^n}$.
• 2.4.A.4 The value of an exponential expression involving an exponential unit fraction, such as $b^{(1/k)}$ where $k$ is a natural number, is the $k$th root of $b$, when it exists.

Learning Objective 2.5.A: Construct a model for situations involving proportional output values over equal-length input-value intervals.

• 2.5.A.1 Exponential functions model growth patterns where successive output values over equal-length input-value intervals are proportional. When the input values are whole numbers, exponential functions model situations of repeated multiplication of a constant to an initial value.
• 2.5.A.2 A constant may need to be added to the dependent variable values of a data set to reveal a proportional growth pattern.
• 2.5.A.3 An exponential function model can be constructed from an appropriate ratio and initial value or from two input-output pairs. The initial value and the base can be found by solving a system of equations resulting from the two input-output pairs.
• 2.5.A.4 Exponential function models can be constructed by applying transformations to $f(x) = ab^x$ based on characteristics of a contextual scenario or data set.
• 2.5.A.5 Exponential function models can be constructed for a data set with technology using exponential regressions.
• 2.5.A.6 The natural base $e$, which is approximately $2.718$, is often used as the base in exponential functions that model contextual scenarios.

Learning Objective 2.6.A: Construct linear, quadratic, and exponential models based on a data set.

• 2.6.A.1 Two variables in a data set that demonstrate a slightly changing rate of change can be modeled by linear, quadratic, and exponential function models.
• 2.6.A.2 Models can be compared based on contextual clues and applicability to determine which model is most appropriate.

Learning Objective 2.6.B: Validate a model constructed from a data set.

• 2.6.B.1 A model is justified as appropriate for a data set if the graph of the residuals of a regression, the residual plot, appear without pattern.
• 2.6.B.2 The difference between the predicted and actual values is the error in the model. Depending on the data set and context, it may be more appropriate to have an underestimate or overestimate for any given interval.

Learning Objective 2.7.A: Evaluate the composition of two or more functions for given values.

• 2.7.A.1 If $f$ and $g$ are functions, the composite function $f \circ g$ maps a set of input values to a set of output values such that the output values of $g$ are used as input values of $f$. For this reason, the domain of the composite function is restricted to those input values of $g$ for which the corresponding output value is in the domain of $f$. $(f \circ g)(x)$ can also be represented as $f(g(x))$.
• 2.7.A.2 Values for the composite function $f \circ g$ can be calculated or estimated from the graphical, numerical, analytical, or verbal representations of $f$ and $g$ by using output values from $g$ as input values for $f$.
• 2.7.A.3 The composition of functions is not commutative; that is, $f \circ g$ and $g \circ f$ are typically different functions; therefore, $f(g(x))$ and $g(f(x))$ are typically different values.
• 2.7.A.4 If the function $f(x) = x$ is composed with any function $g$, the resulting composite function is the same as $g$; that is, $g(f(x)) = f(g(x)) = g(x)$. The function $f(x) = x$ is called the identity function. When composing two functions, the identify function acts in the same way as $0$, the additive identity, when adding two numbers and $1$, the multiplicative identity, when multiplying two numbers.

Learning Objective 2.7.B: Construct a representation of the composition of two or more functions.

• 2.7.B.1 Function composition is useful for relating two quantities that are not directly related by an existing formula.
• 2.7.B.2 When analytic representations of the functions $f$ and $g$ are available, an analytic representation of $f(g(x))$ can be constructed by substituting $g(x)$ for every instance of $x$ in $f$.
• 2.7.B.3 A numerical or graphical representation of $f \circ g$ can often be constructed by calculating or estimating values for $(x, f(g(x)))$.

Learning Objective 2.7.C: Rewrite a given function as a composition of two or more functions.

• 2.7.C.1 Functions given analytically can often be decomposed into less complicated functions. When properly decomposed, the variable in one function should replace each instance of the function with which it was composed.
• 2.7.C.2 An additive transformation of a function, $f$, that results in vertical and horizontal translations can be understood as the composition of $g(x) = x + k$ with $f$.
• 2.7.C.3 A multiplicative transformation of a function, $f$, that results in vertical and horizontal dilations can be understood as the composition of $g(x) = kx$ with $f$.

Learning Objective 2.8.A: Determine the input-output pairs of the inverse of a function.

• 2.8.A.1 On a specified domain, a function, $f$, has an inverse function, or is invertible, if each output value of $f$ is mapped from a unique input value. The domain of a function may be restricted in many ways to make the function invertible.
• 2.8.A.2 An inverse function can be thought of as a reverse mapping of the function. An inverse function, $f^{-1}$, maps the output values of a function, $f$, on its invertible domain to their corresponding input values; that is, if $f(a) = b$, then $f^{-1}(b) = a$. Alternately, on its invertible domain, if a function consists of input-output pairs $(a, b)$, then the inverse function consists of input-output pairs $(b, a)$.

Learning Objective 2.8.B: Determine the inverse of a function on an invertible domain.

• 2.8.B.1 The composition of a function, $f$, and its inverse function, $f^{-1}$, is the identity function; that is, $f\left(f^{-1}(x)\right) = f^{-1}(f(x)) = x$.
• 2.8.B.2 On a function's invertible domain, the function's range and domain are the inverse function's domain and range, respectively. The inverse of the table of values of $y = f(x)$ can be found by reversing the input-output pairs; that is, $(a, b)$ corresponds to $(b, a)$.
• 2.8.B.3 The inverse of the graph of the function $y = f(x)$ can be found by reversing the roles of the $x$- and $y$-axes; that is, by reflecting the graph of the function over the graph of the identity function $h(x) = x$.
• 2.8.B.4 The inverse of the function can be found by determining the inverse operations to reverse the mapping. One method for finding the inverse of the function $f$ is reversing the roles of $x$ and $y$ in the equation $y = f(x)$, then solving for $y = f^{-1}(x)$.
• 2.8.B.5 In addition to limiting the domain of a function to obtain an inverse function, contextual restrictions may also limit the applicability of an inverse function.

Learning Objective 2.9.A: Evaluate logarithmic expressions.

• 2.9.A.1 The logarithmic expression $\log_b c$ is equal to, or represents, the value that the base $b$ must be exponentially raised to in order to obtain the value $c$. That is, $\log_b c = a$ if and only if $b^a = c$, where $a$ and $c$ are constants, $b > 0$, and $b \neq 1$. (when the base of a logarithmic expression is not specified, it is understood as the common logarithm with a base of $10$)
• 2.9.A.2 The values of some logarithmic expressions are readily accessible through basic arithmetic while other values can be estimated through the use of technology.
• 2.9.A.3 On a logarithmic scale, each unit represents a multiplicative change of the base of the logarithm. For example, on a standard scale, the units might be $0, \; 1, \; 2, \; \ldots$, while on a logarithmic scale, using logarithm base $10$, the units might be $10^0, \; 10^1, \; 10^2, \; \ldots$.

Learning Objective 2.10.A: Construct representations of the inverse of an exponential function with an initial value of 1.

• 2.10.A.1 The general form of a logarithmic function is $f(x) = a\log_b x$, with base $b$, where $b > 0$, $b \neq 1$, and $a \neq 0$.
• 2.10.A.2 The way in which input and output values vary together have an inverse relationship in exponential and logarithmic functions. Output values of general-form exponential functions change proportionately as input values increase in equal-length intervals. However, input values of general-form logarithmic functions change proportionately as output values increase in equal-length intervals. Alternately, exponential growth is characterized by output values changing multiplicatively as input values change additively, whereas logarithmic growth is characterized by output values changing additively as input values change multiplicatively.
• 2.10.A.3 $f(x) = \log_b x$ and $g(x) = b^x$, where $b > 0$ and $b \neq 1$, are inverse functions. That is, $g(f(x)) = f(g(x)) = x$.
• 2.10.A.4 The graph of the logarithmic function $f(x) = \log_b x$, where $b > 0$ and $b \neq 1$, is a reflection of the graph of the exponential function $g(x) = b^x$, where $b > 0$ and $b \neq 1$, over the graph of the identity function $h(x) = x$.
• 2.10.A.5 If $(s, \; t)$ is an ordered pair of the exponential function $g(x) = b^x$, where $b > 0$ and $b \neq 1$, then $(t, \; s)$ is an ordered pair of the logarithmic function $f(x) = \log_b x$, where $b > 0$ and $b \neq 1$.

Learning Objective 2.11.A: Identify key characteristics of logarithmic functions.

• 2.11.A.1 The domain of a logarithmic function in general form is any real number greater than zero, and its range is all real numbers.
• 2.11.A.2 Because logarithmic functions are inverses of exponential functions, logarithmic functions are also always increasing or always decreasing, and their graphs are either always concave up or always concave down. Consequently, logarithmic functions do not have extrema except on a closed interval, and their graphs do not have points of inflection.
• 2.11.A.3 The additive transformation function $g(x) = f(x + k)$, where $k \neq 0$, of a logarithmic function $f$ in general form does not have input values that are proportional over equal-length output-value intervals. However, if the input values of the additive transformation function, $g(x) = f(x + k)$, of any function $f$ are proportional over equal-length output value intervals, then $f$ is logarithmic.
• 2.11.A.4 With their limited domain, logarithmic functions in general form are vertically asymptotic to $x = 0$, with an end behavior that is unbounded. That is, for a logarithmic function in general form, $\lim\limits_{x \to 0^+} a\log_b x = \pm\infty$ and $\lim\limits_{x \to \infty} a\log_b x = \pm\infty$.

Learning Objective 2.12.A: Rewrite logarithmic expressions in equivalent forms.

• 2.12.A.1 The product property for logarithms states that $\log_b(xy) = \log_b x + \log_b y$. Graphically, this property implies that every horizontal dilation of a logarithmic function, $f(x) = \log_b(kx)$, is equivalent to a vertical translation, $f(x) = \log_b(kx) = \log_b k + \log_b x = a + \log_b x$, where $a = \log_b k$.
• 2.12.A.2 The power property for logarithms states that $\log_b x^n = n\log_b x$. Graphically, this property implies that raising the input of a logarithmic function to a power, $f(x) = \log_b x^k$, results in a vertical dilation, $f(x) = \log_b x^k = k\log_b x$.
• 2.12.A.3 The change of base property for logarithms states that $\log_b x = \dfrac{\log_a x}{\log_a b}$, where $a > 0$ and $a \neq 1$. This implies that all logarithmic functions are vertical dilations of each other.
• 2.12.A.4 The function $f(x) = \ln x$ is a logarithmic function with the natural base $e$; that is, $\ln x = \log_e x$.

Learning Objective 2.13.A: Solve exponential and logarithmic equations and inequalities.

• 2.13.A.1 Properties of exponents, properties of logarithms, and the inverse relationship between exponential and logarithmic functions can be used to solve equations and inequalities involving exponents and logarithms.
• 2.13.A.2 When solving exponential and logarithmic equations found through analytical or graphical methods, the results should be examined for extraneous solutions precluded by the mathematical or contextual limitations.
• 2.13.A.3 Logarithms can be used to rewrite expressions involving exponential functions in different ways that may reveal helpful information. Specifically, $b^x = c^{(\log_c b)(x)}$.

Learning Objective 2.13.B: Construct the inverse function for exponential and logarithmic functions.

• 2.13.B.1 The function $f(x) = ab^{(x-h)} + k$ is a combination of additive transformations of an exponential function in general form. The inverse of $y = f(x)$ can be found by determining the inverse operations to reverse the mapping.
• 2.13.B.2 The function $f(x) = a\log_b(x - h) + k$ is a combination of additive transformations of a logarithmic function in general form. The inverse of $y = f(x)$ can be found by determining the inverse operations to reverse the mapping.

Learning Objective 2.14.A: Construct a logarithmic function model.

• 2.14.A.1 Logarithmic functions are inverses of exponential functions and can be used to model situations involving proportional growth, or repeated multiplication, where the input values change proportionally over equal-length output-value intervals. Alternately, if the output value is a whole number, it indicates how many times the initial value has been multiplied by the proportion.
• 2.14.A.2 A logarithmic function model can be constructed from an appropriate proportion and a real zero or from two input-output pairs.
• 2.14.A.3 Logarithmic function models can be constructed by applying transformations to $f(x) = a\log_b x$ based on characteristics of a context or data set.
• 2.14.A.4 Logarithmic function models can be constructed for a data set with technology using logarithmic regressions.
• 2.14.A.5 The natural logarithm function is often useful in modeling real-world phenomena.
• 2.14.A.6 Logarithmic function models can be used to predict values for the dependent variable.

Learning Objective 2.15.A: Determine if an exponential model is appropriate by examining a semi-log plot of a data set.

• 2.15.A.1 In a semi-log plot, one of the axes is logarithmically scaled. When the $y$-axis of a semi-log plot is logarithmically scaled, data or functions that demonstrate exponential characteristics will appear linear.
• 2.15.A.2 An advantage of semi-log plots is that a constant never needs to be added to the dependent variable values to reveal that an exponential model is appropriate.

Learning Objective 2.15.B: Construct the linearization of exponential data.

• 2.15.B.1 Techniques used to model linear functions can be applied to a semi-log graph.
• 2.15.B.2 For an exponential model of the form $y = ab^x$, the corresponding linear model for the semi-log plot is $y = (\log_n b)x + \log_n a$, where $n > 0$ and $n \neq 1$. Specifically, the linear rate of change is $\log_n b$, and the initial linear value is $\log_n a$.