Inverse Functions and their Representations

Definition A function is a set of ordered pairs with no two first elements alike. f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) } But ... what if we reverse the order of the pairs? This is also a function ... it is the inverse function f -1(x) = { (x,y) : (2, 3), (4, 1), (6, 7), (12, 9) }

Example Consider an element of an electrical circuit which increases its resistance as a function of temperature. T = Temp R = Resistance -20 50 150 20 250 40 350 R = f(T)

Now we would say that g(R) and f(T) are inverse functions Example We could also take the view that we wish to determine T, temperature as a function of R, resistance. R = Resistance T = Temp 50 -20 150 250 20 350 40 T = g(R) Now we would say that g(R) and f(T) are inverse functions

Terminology If R = f(T) ... resistance is a function of temperature, Then T = f-1(R) ... temperature is the inverse function of resistance. f-1(R) is read "f-inverse of R“ is not an exponent it does not mean reciprocal

Finding the Inverse

Composition of Inverse Functions Consider f(3) = 27   and   f -1(27) = 3 Thus, f(f -1(27)) = 27 and f -1(f(3)) = 3 In general   f(f -1(n)) = n   and f -1(f(n)) = n (assuming both f and f -1 are defined for n)

Domain and Range The domain of f is the range of f -1 The range of f is the domain of f -1 Thus ... we may be required to restrict the domain of f so that f -1 is a function

Linear Functions Parent Function f(x) = x Other Forms: f(x) = mx + b f(x) = b + ax y – y1 = m(x – x1) Ax + By = C Characteristics Algebra I Domain & Range: Zero: x-intercept: y-intercept: Algebra II Increasing/Decreasing: End Behavior: Table As we explore each function in more detail, we will be using multiple representations to discuss each parent function of the function families. We will develop the characteristics of each function as they are addressed in Algebra I and Algebra II and view a graphical and tabular representation of each parent function. Linear functions have many algebraic representations, as shown in the upper left-hand box. In Algebra I, students begin the exploration of domain and range and zeros. The relationship between the zero of a linear function and the solution of a linear equation is an important concept for students to understand. An exploration of intercepts allows students to obtain a deeper recognition of the characteristics of linear functions. In Algebra II, students will discuss the intervals over which functions are increasing and/or decreasing and discuss end behavior.

Linear Functions f(x) = x Parent Function Characteristics Table Other Forms: f(x) = mx + b f(x) = b + ax y – y1 = m(x – x1) Ax + By = C Characteristics Algebra I Domain & Range: {all real numbers} Zero: x=0 x-intercept: (0, 0) y-intercept: (0, 0) Algebra II Increasing/Decreasing: f(x) is increasing over the interval {all real numbers} End Behavior: As x approaches + ∞, f(x) approaches + ∞. As x approaches - ∞, f(x) approaches - ∞. Table As we explore each function in more detail, we will be using multiple representations to discuss each parent function of the function families. We will develop the characteristics of each function as they are addressed in Algebra I and Algebra II and view a graphical and tabular representation of each parent function. Linear functions have many algebraic representations, as shown in the upper left-hand box. In Algebra I, students begin the exploration of domain and range and zeros. The relationship between the zero of a linear function and the solution of a linear equation is an important concept for students to understand. An exploration of intercepts allows students to obtain a deeper recognition of the characteristics of linear functions. In Algebra II, students will discuss the intervals over which functions are increasing and/or decreasing and discuss end behavior.

Absolute Value Functions Parent Function f(x) = |x| Other Forms: f(x) = a|x - h| + k Characteristics Algebra II Domain: Range: Zeros: x-intercept: y-intercept: Increasing/Decreasing: End Behavior: Table of Values Review the absolute value function and consider how students in Algebra II might attempt to identify the characteristics of this function given what they know about linear functions. Take a moment to jot down the characteristics of the absolute value function and complete the table of values shown and then compare your answers with someone seated near you. How do the ideas deemed necessary about this parent function compare to your discussion of functions during the Placemat consensus activity?

Absolute Value Functions Parent Function f(x) = |x| Other Forms: f(x) = a|x - h| + k Characteristics Algebra II Domain: {all real numbers} Range: {f(x)| f(x) > 0} Zeros: x=0 x-intercept: (0, 0), y-intercept: (0, 0) Increasing/Decreasing: Dec: {x| -∞ < x < 0} Inc: {x| 0 < x < ∞} End Behavior: As x approaches + ∞, f(x) approaches + ∞. As x approaches - ∞, f(x) approaches + ∞. Table of Values Compare your answers about the characteristics of the absolute value function to those shown. Why is it important for students to recognize these characteristics about functions?

Quadratic Functions Parent Function Characteristics Table Other Forms: Algebra I Domain: Range: Zeros: x-intercept: y-intercept: Algebra II Increasing/Decreasing: End Behavior: Table Quadratic functions can be the first formal introduction to non-linear relationships that students see in mathematics. Students should be given the opportunity to explore many real-world situations that model quadratic functions such as projectile motion, in order to have concrete examples that relate to these more abstract ideas. 13 13

Quadratic Functions Parent Function Characteristics Table Other Forms: Algebra I Domain: {all real numbers} Range: {f(x)| f(x) > 0} Zeros: x=0 x-intercept: (0, 0), y-intercept: (0, 0) Algebra II Increasing/Decreasing: Dec: {x| -∞ < x < 0} Inc: {x| 0 < x < ∞} End Behavior: As x approaches - ∞, f(x) approaches + ∞. As x approaches + ∞, f(x) approaches + ∞. Table Quadratic functions can be the first formal introduction to non-linear relationships that students see in mathematics. Students should be given the opportunity to explore many real-world situations that model quadratic functions such as projectile motion, in order to have concrete examples that relate to these more abstract ideas. 14 14

Square Root Functions Parent Function Characteristics Table Other Forms: Characteristics Algebra II Domain: Range: Zeros: x-intercept: y-intercept: Increasing/Decreasing: End Behavior: Table The square root function allows students to begin to think about graphical inverses. A simple activity using patty paper and tables of values can assist student s in recognizing the inverse relationship between the quadratic and square root function. This is the first function students may encounter where the domain is restricted. For the parent function, x is restricted to be greater than or equal to zero, which in turn creates a restriction on the range of the function. Students may informally note that the square root function operates similarly to a ray in Geometry, in that it has an endpoint and goes off to infinity in only one direction (be sure to point out that the square root function is not linear, like a ray). Similarly, we can only discuss end behavior of the square root function for only one part of the graph. 15 15

Square Root Functions Parent Function Characteristics Table Other Forms: Characteristics Algebra II Domain: {x| x > 0 } Range: {f(x)| f(x) > 0} Zeros: x=0 x-intercept: (0, 0) y-intercept: (0, 0) Increasing/Decreasing: Increasing on {x| 0 < x < ∞} End Behavior: As x approaches + ∞, f(x) approaches + ∞. Table The square root function allows students to begin to think about graphical inverses. A simple activity using patty paper and tables of values can assist student s in recognizing the inverse relationship between the quadratic and square root function. This is the first function students may encounter where the domain is restricted. For the parent function, x is restricted to be greater than or equal to zero, which in turn creates a restriction on the range of the function. Students may informally note that the square root function operates similarly to a ray in Geometry, in that it has an endpoint and goes off to infinity in only one direction (be sure to point out that the square root function is not linear, like a ray). Similarly, we can only discuss end behavior of the square root function for only one part of the graph. 16 16

Cube Root Functions Parent Function Characteristics Table Other Forms: Algebra II Domain: Range: Zeros: x-intercept: y-intercept: Increasing Interval: End Behavior: Table Students should be encouraged to compare the characteristics of the cube root function to the square root function. Why is the domain of a square root function restricted, but not the domain of a cube root function? Why is the range also the set of all real numbers? As students explore these ideas, their prior knowledge of evaluating square and cube roots will be activated. When we naturally encourage students to make these connections, the understanding of algebraic functions deepens. 17 17

Cube Root Functions Parent Function Characteristics Table Other Forms: Algebra II Domain: {all real numbers } Range: {all real numbers } Zeros: x=0 x-intercept: (0, 0) y-intercept: (0, 0) Increasing Interval: {all real numbers} End Behavior: As x approaches - ∞, f(x) approaches - ∞; As x approaches + ∞, f(x) approaches + ∞. Table Students should be encouraged to compare the characteristics of the cube root function to the square root function. Why is the domain of a square root function restricted, but not the domain of a cube root function? Why is the range also the set of all real numbers? As students explore these ideas, their prior knowledge of evaluating square and cube roots will be activated. When we naturally encourage students to make these connections, the understanding of algebraic functions deepens. 18 18

Rational Functions Parent Function Characteristics Table Other Forms: where a(x) and b(x) are polynomial functions Characteristics Algebra II Domain: Range: Zeros: x-intercept & y-intercept: Increasing/Decreasing: End Behavior: Asymptotes: Table The parent function y = 1/x possesses two asymptotes. A vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The parameters of the parent function can be adjusted to allow students to explore the affect on asymptotes. 19 19

Rational Functions Parent Function Characteristics Table Other Forms: where a(x) and b(x) are polynomial functions Characteristics Algebra II Domain: {x| x<0} U {x| x>0} Range: {f(x)| f(x) < 0} U {f(x)| f(x) > 0} Zeros: none x-intercept & y-intercept: none Decreasing: {x| -∞ < x < 0} U {x| 0 < x < ∞} End Behavior: As x approaches - ∞, f(x) approaches 0; as x approaches + ∞, f(x) approaches 0. Asymptotes: x = 0, y = 0 Table The parent function y = 1/x possesses two asymptotes. A vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The parameters of the parent function can be adjusted to allow students to explore the affect on asymptotes. 20 20

Exponential Functions Parent Function Other Forms: Characteristics (f(x) = 2x) Algebra II Domain: Range: Zeros: x-intercepts: y-intercepts: Asymptote: End Behavior: Table Exponential Functions become important as the variable moves from the base to the exponent. This is where asymptotes become obvious as students work through values that cannot exist. … this topic requires preparation in the handling of negative exponents and their use in fraction form (and not negative numbers) Students will also experience ranges that become very large very fast (or very small) as x-values increase slowly.

Exponential Functions Parent Function Other Forms: Characteristics (f(x) = 2x) Algebra II Domain: {all real numbers} Range: {f(x)| f(x) > 0} Zeros: none x-intercepts: none y-intercepts: (0, 1) Asymptote: y = 0 End Behavior: As x approaches ∞, f(x) approaches + ∞. As x approaches - ∞, f(x) approaches 0. Table Exponential Functions become important as the variable moves from the base to the exponent. This is where asymptotes become obvious as students work through values that cannot exist. … this topic requires preparation in the handling of negative exponents and their use in fraction form (and not negative numbers) Students will also experience ranges that become very large very fast (or very small) as x-values increase slowly.

Logarithmic Functions Parent Function f(x) = logb x, b > 0, b 1 Characteristics (f(x) = log x) Algebra II Domain: Range: Zeros: x-intercepts: y-intercepts: Asymptotes: End Behavior: Table Logarithmic functions are the inverses of exponential functions and can best be introduced through this relationship. Students should notice that x- and y-values changes in the table of values and likewise domain and range characteristics change in the same manner. Caution needs to occur to avoid students from thinking that all the characteristics of the exponential function are switched on the logarithmic functions which is why students need to be prepared for the analysis of these functions through their experiences with the polynomial and square root functions. Continued work on logarithms may distance the student from the graphical connection to it’s use as an inverse to the exponential function, and thus, teachers should continually help students’ recall their prior work in this context.

Logarithmic Functions Parent Function f(x) = logb x, b > 0, b 1 Characteristics (f(x) = log x) Algebra II Domain: {x| x > 0} Range: {all real numbers} Zeros: x=1 x-intercepts: (1, 0) y-intercepts: none Asymptotes: x = 0 End Behavior: As x approaches ∞, y approaches + ∞. Table Logarithmic functions are the inverses of exponential functions and can best be introduced through this relationship. Students should notice that x- and y-values changes in the table of values and likewise domain and range characteristics change in the same manner. Caution needs to occur to avoid students from thinking that all the characteristics of the exponential function are switched on the logarithmic functions which is why students need to be prepared for the analysis of these functions through their experiences with the polynomial and square root functions. Continued work on logarithms may distance the student from the graphical connection to it’s use as an inverse to the exponential function, and thus, teachers should continually help students’ recall their prior work in this context.