Question

Danika concludes that the following functions are inverses of each other because f(g(x)) = x. Do you agree with Danika? Explain your reasoning.
f(x) = |x|
g(x) = –x

Answer 1

Let's take functions fff and ggg for example: f(x)=\dfrac{x+1}{3}f(x)=​3​​x+1​​f, left parenthesis, x, right parenthesis, equals, start fraction, x, plus, 1, divided by, 3, end fraction and g(x)=3x-1g(x)=3x−1g, left parenthesis, x, right parenthesis, equals, 3, x, minus, 1.Notice how f(5)=2f(5)=2f, left parenthesis, 5, right parenthesis, equals, 2 and g(2)=5g(2)=5g, left parenthesis, 2, right parenthesis, equals, 5. [Please show me the calculations]f(5)f, left parenthesis, 5, right parenthesis55ff\begin{aligned}f(x)&=\dfrac{x+1}{3}\\ \\ f(\blueD{5})&=\dfrac{\blueD5+1}{3}\\ \\ &=\goldD2 \end{aligned}g(2)g, left parenthesis, 2, right parenthesis22gg\begin{aligned}g(x)&=3x-1\\ \\ g(\goldD{2})&=3(\goldD2)-1\\ \\ &=\blueD5 \end{aligned}\blueD{5}5ff\goldD{2}2gg\blueD{5}5