Question

Consider he function f(x)=2x-4 a.find the inverse of (x) and name it g(x) . Show your work

B. Use function composition to show that f(x) and g(x) are inverses of each other .show work

Answer 1

`A.\\f(x)=2x-4\to y=2x-4\\\\\text{change x to y and y to x}\\\\x=2y-4\\\\\text{solve for y}\\\\2y-4=x\qquad|+4\\\\2y=x+4\qquad|:2\\\\y=(1)/(2)x+2\\\\f^(-1)(x)=g(x)=(1)/(2)x+2\\B.\\\text{Verifying if two functions are inverses of each other is a simple two-step process.}\\1.\ \text{Plug}\ f(x)\ \text{into}\ g(x)\ \text{and simplify}\\2.\  \text{Plug}\ g(x)\ \text{into}\ f(x)\ \text{and simplify}\\\\If\ g[f(x)]=f[g(x)]=x,\ then\ functions\ are\ inverses\ of\ each\\\ other.`

`f(x)=2x-4,\ g(x)=(1)/(2)x+2\\\\g[f(x)]=(1)/(2)(2x-4)+2=(1)/(2)(2x)+(1)/(2)(-4)+2=x-2+2=x\\\\f[g(x)]=2\left((1)/(2)x+2\right)-4=2\left((1)/(2)x\right)+2(2)-4=x+4-4=x\\\\\text{Since the results above came out OK, both}\ x,\text{ then we can claim}\\\text{that functions f (x) and g (x) are inverses of each other.}`