If g(f(x))= 2x+1, f(x)= (1/4)x-1, then what does g(x) equal? The answer is supposed to be 8x+9. Can anyone explain to me clearly step by step how to do this? Thanks!
Hello, woahitzme!
$\displaystyle \text{If }g(f(x))\:=\: 2x+1,\;\; f(x)\:=\:\tfrac{1}{4}x-1$
. . $\displaystyle \text{then what does }g(x)\text{ equal?}$
$\displaystyle \text{The answer is supposed to be: }\,8x+9.$
We are told that: .$\displaystyle f(x) \:=\:\tfrac{1}{4}x-1$
Then: .$\displaystyle g\left(\tfrac{1}{4}x-1\right) \:=\:2x+1$
Assume that $\displaystyle g(x)$ is a linear function: .$\displaystyle g(x) \:=\:ax+b$
Then: .$\displaystyle g\left(\tfrac{1}{4}x-1\right) \:=\:a\left(\tfrac{1}{4}x-1\right) + b \;=\;2x + 1$
We have: .$\displaystyle \tfrac{a}{4}x - a + b \:=\:2x+1 \quad\Rightarrow\quad \tfrac{a}{4}x + (b-a) \:=\:2x+1$
Equate coefficients: .$\displaystyle \begin{array}{ccc}\dfrac{a}{4} &=& 2 \\ b-a &=& 1 \end{array}$
Hence: .$\displaystyle a \,=\,8,\;b\,=\,9$
Therefore: .$\displaystyle g(x) \:=\:8x + 9$