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Chapter 2: Problem 95

If \(f(x)=3 x\) and \(g(x)=x+5,\) find \((f \circ g)^{-1}(x)\) and \(\left(g^{-1} \circ f^{-1}\right)(x)\)

Short Answer

Expert verified
\((f \circ g)^{-1}(x) = (x-15)/3\) and \((g^{-1} \circ f^{-1})(x) = (x/3)-5\)

Step by step solution

01

Find the Composition of f and g

First calculate the composition of the two functions \(f\) and \(g\) which is denoted as \(f \circ g\). This is done by substituting \(g(x)\) into the function \(f\). Thus, \(f(g(x))=3(g(x))=3(x+5)=3x+15.\)
02

Find the Inverse of the Composition

Next, find \((f \circ g)^{-1}(x)\), the inverse of the composition of \(f\) and \(g\). Firstly, replace \(f(g(x))\) with \(y\), so the equation becomes \(y = 3x + 15\). Then, switch the roles of \(x\) and \(y\) to get \(x = 3y + 15\). Finally, solve the equation for \(y\) which gives \((f \circ g)^{-1}(x) = (x-15)/3\)
03

Find the Inverses of f and g

Now find the inverse of \(f(x)\) and \(g(x)\) separately. For \(f(x)=3x\), replace \(f(x)\) with \(y\), so the equation becomes \(y = 3x\). Swap \(x\) and \(y\) to get \(x = 3y\), and solve for \(y\) to obtain \(f^{-1}(x) = x/3\). Similarly, for \(g(x) = x+5\), the inverse \(g^{-1}(x) = x - 5 \).
04

Calculate the Composition of the Inverses

Finally, calculate the composition of the inverses as \((g^{-1} \circ f^{-1})(x)\). Insert the inverse of \(f(x)\) into \(g^{-1}(x)\), to obtain \(g^{-1}(f^{-1}(x)) = (x/3)-5\)

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Most popular questions from this chapter

Explain how to find the difference quotient of a function \(f\) \(\frac{f(x+h)-f(x)}{h},\) if an equation for \(f\) is given.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I found the inverse of \(f(x)=5 x-4\) in my head: The reverse of multiplying by 5 and subtracting 4 is adding 4 and dividing by \(5,\) so \(f^{-1}(x)=\frac{x+4}{5}\).

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=\sqrt[3]{2-x} $$

a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1, g(x)=f(2 x), h(x)=f(3 x),\) and \(k(x)=f(4 x)\) in the same viewing rectangle. c. Describe the relationship among the graphs of \(f, g, h\) and \(k,\) with emphasis on different values of \(x\) for points on all four graphs that give the same \(y\) -coordinate. d. Generalize by describing the relationship between the graph of \(f\) and the graph of \(g,\) where \(g(x)=f(c x)\) for \(c>1\) e. Try out your generalization by sketching the graphs of \(f(c x)\) for \(c=1, c=2, c=3,\) and \(c=4\) for a function of your choice.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=3 x,\) then \(f^{-1}(x)=\frac{1}{3 x}\)
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