Section 1
Composite and Inverse Functions
Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.$$\begin{array}{lllll}\text { inverse } & \text { domain } & \text { horizontal } & x & f(x) \\\text { vertical } & \text { range } & \text { one-to-one } & \text { composition } & y\end{array}.$$When one variable is a function of another variable which in turn is a function of a third variable, we describe the relationship among such functions as a of ______ functions.
Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.$$\begin{array}{lllll}\text { inverse } & \text { domain } & \text { horizontal } & x & f(x) \\\text { vertical } & \text { range } & \text { one-to-one } & \text { composition } & y\end{array}.$$A function is a __________ function if each element in the range corresponds to exactly one element in the domain.
Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.$$\begin{array}{lllll}\text { inverse } & \text { domain } & \text { horizontal } & x & f(x) \\\text { vertical } & \text { range } & \text { one-to-one } & \text { composition } & y\end{array}.$$The _____ line test states that, if a ____ line can be drawn so that it intersects a graph at more than one point, then the graph is not the graph of a function.
Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.$$\begin{array}{lllll}\text { inverse } & \text { domain } & \text { horizontal } & x & f(x) \\\text { vertical } & \text { range } & \text { one-to-one } & \text { composition } & y\end{array}.$$The _____ line test states that, if a line ____ can be drawn so that it intersects the graph of a function at more than one point, the function is not a one-to-one function.
Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.$$\begin{array}{lllll}\text { inverse } & \text { domain } & \text { horizontal } & x & f(x) \\\text { vertical } & \text { range } & \text { one-to-one } & \text { composition } & y\end{array}.$$If a one-to-one function has ordered pairs of the form $(x, y)$, the ____ function is a one-to-one function with ordered pairs of the form $(y, x)$.
Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.$$\begin{array}{lllll}\text { inverse } & \text { domain } & \text { horizontal } & x & f(x) \\\text { vertical } & \text { range } & \text { one-to-one } & \text { composition } & y\end{array}.$$For a one-to-one function $f(x)$ and its inverse function $f^{-1}(x)$, the ____ domain of $f(x)$ is the of $f^{-1}(x)$.
Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.$$\begin{array}{lllll}\text { inverse } & \text { domain } & \text { horizontal } & x & f(x) \\\text { vertical } & \text { range } & \text { one-to-one } & \text { composition } & y\end{array}.$$For a one-to-one function $f(x)$ and its inverse function $f^{-1}(x)$, the range of $f(x)$ is the ______ of $f^{-1}(x)$.
Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.$$\begin{array}{lllll}\text { inverse } & \text { domain } & \text { horizontal } & x & f(x) \\\text { vertical } & \text { range } & \text { one-to-one } & \text { composition } & y\end{array}.$$For any one-to-one function $f(x)$ and its inverse $f^{-1}(x)$, the range of $f(x)$ is the _____ and $\left(f^{-1} \circ f\right)(x)=$ ____.
For each pair of functions, determine a) $(f \circ g)(x), b)(f \circ g)(4), \mathbf{c})(g \circ f)(x)$, and d) $(g \circ f)(4)$.$f(x)=x+4, g(x)=2 x-3$
For each pair of functions, determine a) $(f \circ g)(x), b)(f \circ g)(4), \mathbf{c})(g \circ f)(x)$, and d) $(g \circ f)(4)$.$f(x)=3 x-2, g(x)=x+1$
For each pair of functions, determine a) $(f \circ g)(x), b)(f \circ g)(4), \mathbf{c})(g \circ f)(x)$, and d) $(g \circ f)(4)$.$f(x)=x+3, g(x)=x^{2}+x-4$
For each pair of functions, determine a) $(f \circ g)(x), b)(f \circ g)(4), \mathbf{c})(g \circ f)(x)$, and d) $(g \circ f)(4)$.$f(x)=x+2, g(x)=x^{2}+4 x-2$
For each pair of functions, determine a) $(f \circ g)(x), b)(f \circ g)(4), \mathbf{c})(g \circ f)(x)$, and d) $(g \circ f)(4)$.$f(x)=\frac{1}{x^{\prime}} g(x)=2 x+3$
For each pair of functions, determine a) $(f \circ g)(x), b)(f \circ g)(4), \mathbf{c})(g \circ f)(x)$, and d) $(g \circ f)(4)$.$f(x)=3 x+1, g(x)=\frac{3}{x}$
For each pair of functions, determine a) $(f \circ g)(x), b)(f \circ g)(4), \mathbf{c})(g \circ f)(x)$, and d) $(g \circ f)(4)$.$f(x)=\frac{2}{x^{\prime}}, g(x)=x^{2}+1$
For each pair of functions, determine a) $(f \circ g)(x), b)(f \circ g)(4), \mathbf{c})(g \circ f)(x)$, and d) $(g \circ f)(4)$.$f(x)=x^{2}-5, g(x)=\frac{4}{x}$
For each pair of functions, determine a) $(f \circ g)(x), b)(f \circ g)(4), \mathbf{c})(g \circ f)(x)$, and d) $(g \circ f)(4)$.$f(x)=x^{2}+1, g(x)=x^{2}+5$
For each pair of functions, determine a) $(f \circ g)(x), b)(f \circ g)(4), \mathbf{c})(g \circ f)(x)$, and d) $(g \circ f)(4)$.$f(x)=x^{2}-4, g(x)=x^{2}+3$
For each pair of functions, determine a) $(f \circ g)(x), b)(f \circ g)(4), \mathbf{c})(g \circ f)(x)$, and d) $(g \circ f)(4)$.$f(x)=x-4, g(x)=\sqrt{x+5}, x \geq-5$
For each pair of functions, determine a) $(f \circ g)(x), b)(f \circ g)(4), \mathbf{c})(g \circ f)(x)$, and d) $(g \circ f)(4)$.$f(x)=\sqrt{x+6}, x \geq-6, g(x)=x+7$
In Exercises 21-42, determine whether each function is a one-to-one function.Arrow Cant copy
Determine whether each function is a one-to-one function.$\{(1,1),(2,2),(3,3),(4,4)\}$
Determine whether each function is a one-to-one function.$\{(1,2),(2,3),(3,4),(4,5)\}$
Determine whether each function is a one-to-one function.$\{(-4,2),(5,3),(0,2),(4,8)\}$
Determine whether each function is a one-to-one function.$\{(0,5),(1,4),(-3,5),(4,2)\}$
Determine whether each function is a one-to-one function.$y=2 x+5$
Determine whether each function is a one-to-one function.$y=3 x-8$
Determine whether each function is a one-to-one function.$y=x^{2}-1$
Determine whether each function is a one-to-one function.$y=-x^{2}+3$
Determine whether each function is a one-to-one function.$y=x^{2}-9, x \geq 0$
Determine whether each function is a one-to-one function.$y=x^{2}-9, x \leq 0$
Determine whether each function is a one-to-one function.$y=x^{2}-2 x-3, x \geq 1$
Determine whether each function is a one-to-one function.$y=x^{2}+4 x-5, x \geq-2$
Determine whether each function is a one-to-one function.$y=\sqrt{x}$
Determine whether each function is a one-to-one function.$y=-\sqrt{x}$
Determine whether each function is a one-to-one function.$y=|x|$
Determine whether each function is a one-to-one function.$y=-|x|$
Determine whether each function is a one-to-one function.$y=\sqrt[3]{x}$
Determine whether each function is a one-to-one function.$y=x^{3}$
In Exercises 43-48, for the given function, determine the domain and range of both $f(x)$ and $f^{-1}(x)$.$\{(4,0),(8,9),(2,7),(-1,6),(-2,4)\}$
For the given function, determine the domain and range of both $f(x)$ and $f^{-1}(x)$.$\left\{(-2,-3),(-4,0),(5,3),(6,2),\left(2, \frac{1}{2}\right)\right\}$
For the given function, determine the domain and range of both $f(x)$ and $f^{-1}(x)$.Graph Can`t Copy
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$f(x)=x+5$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$f(x)=x-4$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$h(x)=4 x$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$k(x)=2 x-7$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$r(x)=|x|$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$t(x)=-|x|$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$t(x)=x^{2}+3$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$m(x)=-x^{2}+x+8$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$g(x)=\frac{1}{x}$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$h(x)=\frac{5}{x}$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$g(x)=x^{3}-6$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$g(x)=x^{3}+9$
$g(x)=\sqrt{x+2}, x \geq-2$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$g(x)=\sqrt{x+2}, x \geq-2$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$f(x)=\sqrt{x}, x \geq 0$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$h(x)=x^{2}-4, x \geq 0$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$f(x)=x^{2}-3, x \geq 0$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$g(x)=\sqrt[3]{x-1}$
For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.$f(x)=\sqrt[3]{x}-2$
For each one-to-one function, a) determine $f^{-1}(x)$ and b) graph $f(x)$ and $f^{-1}(x)$ on the same axes.$f(x)=2 x+8$
For each one-to-one function, a) determine $f^{-1}(x)$ and b) graph $f(x)$ and $f^{-1}(x)$ on the same axes.$f(x)=-3 x+6$
For each one-to-one function, a) determine $f^{-1}(x)$ and b) graph $f(x)$ and $f^{-1}(x)$ on the same axes.$f(x)=\sqrt{x}, x \geq 0$
For each one-to-one function, a) determine $f^{-1}(x)$ and b) graph $f(x)$ and $f^{-1}(x)$ on the same axes.$f(x)=-\sqrt{x}, x \geq 0$
For each one-to-one function, a) determine $f^{-1}(x)$ and b) graph $f(x)$ and $f^{-1}(x)$ on the same axes.$f(x)=\sqrt{x-1}, x \geq 1$
For each one-to-one function, a) determine $f^{-1}(x)$ and b) graph $f(x)$ and $f^{-1}(x)$ on the same axes.$f(x)=\sqrt{x+4}, x \geq-4$
For each one-to-one function, a) determine $f^{-1}(x)$ and b) graph $f(x)$ and $f^{-1}(x)$ on the same axes.$f(x)=\sqrt[3]{x+3}$
For each one-to-one function, a) determine $f^{-1}(x)$ and b) graph $f(x)$ and $f^{-1}(x)$ on the same axes.$f(x)=\sqrt[3]{x}$
For each one-to-one function, a) determine $f^{-1}(x)$ and b) graph $f(x)$ and $f^{-1}(x)$ on the same axes.$f(x)=\frac{1}{x}, x>0$
For each one-to-one function, a) determine $f^{-1}(x)$ and b) graph $f(x)$ and $f^{-1}(x)$ on the same axes.$f(x)=\frac{1}{x}$
For each pair of inverse functions, show that $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$.$f(x)=x-7, f^{-1}(x)=x+7$
For each pair of inverse functions, show that $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$.$f(x)=-2 x, f^{-1}(x)=-\frac{1}{2} x$
For each pair of inverse functions, show that $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$.$f(x)=\frac{1}{2} x+3, f^{-1}(x)=2 x-6$
For each pair of inverse functions, show that $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$.$f(x)=-\frac{1}{3} x+2, f^{-1}(x)=-3 x+6$
For each pair of inverse functions, show that $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$.$f(x)=\sqrt[3]{x+7}, f^{-1}(x)=x^{3}-7$
For each pair of inverse functions, show that $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$.
For each pair of inverse functions, show that $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$.$f(x)=\frac{3}{x^{\prime}} f^{-1}(x)=\frac{3}{x}$
For each pair of inverse functions, show that $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$.$f(x)=-\frac{2}{x^{\prime}}, f^{-1}(x)=-\frac{2}{x}$
For each pair of inverse functions, show that $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$.$f(x)=x^{2}+1, x \geq 0, f^{-1}(x)=\sqrt{x-1}$
For each pair of inverse functions, show that $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$.$f(x)=\sqrt{x+5}, f^{-1}(x)=x^{2}-5, x \geq 0$
The function $f(x)=3 x$ converts yards, $x$, into feet. Determine the inverse function that converts feet into yards. In the inverse function, what do $x$ and $f^{-1}(x)$ represent?
The function $f(x)=12 x$ converts feet, $x$, into inches. Determine the inverse function that converts inches into feet. In the inverse function, what do $x$ and $f^{-1}(x)$ represent?
The function $f(x)=\frac{5}{9}(x-32)$ converts degrees Fahrenheit, $x$, to degrees Celsius. Determine the inverse function that changes degrees Celsius into degrees Fahrenheit.
The function $f(x)=\frac{22}{15} x$ converts miles per hour, $x$, into feet per second. Determine the inverse function that converts feet per second into miles per hour.
In Exercises 91-94, the functions $f(x)$ and $g(x)$ are given. Determine the composition $(g \circ f)(x)$. For the composition function, what does $x$ represent and what does $(g \circ f)(x)$ represent?$f(x)=16 x$ converts pounds, $x$, to ounces. $g(x)=28.35 x$ converts ounces, $x$ to grams.
The functions $f(x)$ and $g(x)$ are given. Determine the composition $(g \circ f)(x)$. For the composition function, what does $x$ represent and what does $(g \circ f)(x)$ represent?$f(x)=2000 x$ converts tons, $x$, to pounds. $g(x)=16 x$ converts pounds, $x$, to ounces.
The functions $f(x)$ and $g(x)$ are given. Determine the composition $(g \circ f)(x)$. For the composition function, what does $x$ represent and what does $(g \circ f)(x)$ represent?
The functions $f(x)$ and $g(x)$ are given. Determine the composition $(g \circ f)(x)$. For the composition function, what does $x$ represent and what does $(g \circ f)(x)$ represent?$f(x)=1760 x$ converts miles, $x$, to yards. $g(x)=0.915 x$ converts yards, $x$, to meters.
Is $(f \circ g)(x)=(g \circ f)(x)$ for all values of $x$ ? Explain and give an example to support your answer.
Consider the functions $f(x)=\sqrt{x+5}, x \geq-5$, and $g(x)=x^{2}-5, x \geq 0 .$a) Show that $(f \circ g)(x)=(g \circ f)(x)$ for $x \geq 0$.b) Explain why we need to stipulate that $x \geq 0$ for part a) to be true.
Consider the functions $f(x)=x^{3}+2$ and $g(x)=\sqrt[3]{x-2} .$a) Show that $(f \circ g)(x)=(g \circ f)(x)$.b) What are the domains of $f(x), g(x),(f \circ g)(x)$, and $(g \circ f)(x) ?$ Explain.
For the function $f(x)=x^{3}, f(2)=2^{3}=8$. Explain why $f^{-1}(8)=2 .$
For the function $f(x)=x^{4}, x>0, f(2)=16$. Explain why $f^{-1}(16)=2$.
a) Does the function $f(x)=|x|$ have an inverse? Explain.b) If the domain is limited to $x \geq 0$, does the function have an inverse? Explain.c) Determine the inverse function of $f(x)=|x|, x \geq 0$.
When a pebble is thrown into a pond, the circle formed by the pebble hitting the water expands with time. The area of the expanding circle may be determined by the formula $A=\pi r^{2}$. The radius, $r$, of the circle, in feet, is a function of time, $t$, in seconds. Suppose that the function is $r(t)=2 t$.a) Determine the radius of the circle at 3 seconds.b) Determine the area of the circle at 3 seconds.c) Express the area as a function of time by determining $A \circ r$.d) Using the function determined in part c), determine the area of the circle at 3 seconds.
The surface area, $S$, of a spherical balloon of radius $r$, in inches, is determined by $S(r)=4 \pi r^{2}$. If the balloon is being blown up at a constant rate by a machine, then the radius of the balloon is a function of time. Suppose that this function is $r(t)=1.2 t$, where $t$ is in seconds.a) Determine the radius of the balloon at 2 seconds.b) Determine the surface area at 2 seconds.c) Express the surface area as a function of time by determining $S \circ r$.d) Using the function determined in part c), determine the surface area after 2 seconds.e) Do your answers in parts b) and d) agree? If not, explain why not.
Consider the function $f(x)=2^{x}$. This is an example of an exponential function, which we will discuss in the next section.a) Graph this function by substituting values for $x$ and determing the corresponding values of $f(x)$.b) Do you think this function has an inverse? Explain your answer.c) Using the graph in part a), draw the inverse function, $f^{-1}(x)$ on the same axes.d) Explain how you obtained the graph of $f^{-1}(x)$
Divide $\left|\frac{-9}{4}\right| \div\left|\frac{-4}{9}\right|$.
Simplify $\frac{\frac{3}{x^{2}}-\frac{2}{x}}{\frac{x}{6}}$.
Solve the formula $\frac{1}{f}=\frac{1}{p}+\frac{1}{q}$ for $p$.
Solve $x^{2}+2 x-10=0$ by completing the square.