1.7 Inverse Functions

Introduction

When a function has an inverse function

Determining whether a function has an inverse function

Recall that $F = g(C) = \frac{9}{5}C + 32$ is the function that takes Celsius temperature inputs and produces the corresponding Fahrenheit temperature outputs.

Show that it is possible to solve the equation $F = \frac{9}{5}C + 32$ for $C$ in terms of $F$ and that doing so results in the equation $C = \frac{5}{9}(F-32)$.

Solve the equation $F = 40 + \frac{1}{4}N$ for $N$ in terms of $F$. Call the resulting function $N = E(F)$.

Explain in words the process or effect of the function $N = E(F)$. What does it take as input? What does it generate as output?

Use the function $E$ that you found in (a.) to compute $j(N) = E(D(N))$. Simplify your result as much as possible. Do likewise for $k(F) = D(E(F))$. What do you notice about these two composite functions $j$ and $k$?

Consider the equations $F = 40 + \frac{1}{4}N$ and $N = 4(F-40)$. Do these equations express different relationships between $F$ and $N$, or do they express the same relationship in two different ways? Explain.

Do the functions $f$ and $g$ defined by Table and Table have corresponding inverse functions? Why or why not?

$x$ | $0$ | $1$ | $2$ | $3$ | $4$

$f(x)$ | $6$ | $4$ | $3$ | $4$ | $6$

$x$ | $0$ | $1$ | $2$ | $3$ | $4$

$g(x)$ | $3$ | $1$ | $4$ | $2$ | $0$

Do the functions $p$ and $q$ defined by Figure and Figure have corresponding inverse functions? Why or why not?

Do the functions $r$ and $s$ defined by

have corresponding inverse functions? If not, use algebraic reasoning to explain why; if so, demonstrate by using algebra to find a formula for the inverse function.

The function $f : S \to S$ given by the table of values below, where $S = \{0, 1, 2, 3, 4 \}$.

$x$ | 0 | 1 | 2 | 3 | 4

$f(x)$ | 1 | 2 | 4 | 3 | 2

The function $g : S \to S$ given by the table of values below, where $S = \{0, 1, 2, 3, 4 \}$.

$x$ | 0 | 1 | 2 | 3 | 4

$g(x)$ | 4 | 0 | 3 | 1 | 2

The function $p$ given by $p(t) = 7 - \frac{3}{5}t$. Assume that the domain and codomain of $p$ are both “all real numbers”.

The function $q$ given by $q(t) = 7 - \frac{3}{5}t^4$. Assume that the domain and codomain of $q$ are both “all real numbers”.

The functions $r$ and $s$ given by the graphs in the figures below. Assume that the graphs show all of the important behavior of the functions and that the apparent trends continue beyond what is pictured.