Answer: a) [tex]\frac{5}{2}x+10=f^{-1}(x)=g(x)[/tex]
Step-by-step explanation:
Since we have given that
[tex]f(x)=\frac{2}{5}x-4[/tex]
a.) Find the inverse of f(x) and name it g(x).
Let f(x) = y
So, it becomes
[tex]y=\frac{2}{5}x-4[/tex]
Switching x to y , we get
[tex]x=\frac{2}{5}y-4[/tex]
[tex]5x=2y-20\\\\5x+20=2y\\\\\frac{5x+20}{2}=y\\\\\frac{5}{2}x+10=y\\\\\frac{5}{2}x+10=f^{-1}(x)=g(x)[/tex]
b) . Use composition to show that f(x) and g(x) are inverses of each other.
[tex]\mathrm{For}\:f=\frac{2}{5}x-4\:\\\\\mathrm{substitute}\:x\:\mathrm{with}\:g\left(x\right)=\frac{5}{2}x+10\\\\=\frac{2}{5}\left(\frac{5}{2}x+10\right)-4\\\\=x[/tex]
Similarly,
[tex]\mathrm{g\left(x\right)=\frac{5}{2}x+10,\:f\left(x\right)=\frac{2}{5}x-4,\:g\left(x\right)\circ \:f\left(x\right)}\\\\\mathrm{For}\:g=\frac{5}{2}x+10\:\mathrm{substitute}\:x\:\mathrm{with}\:f\left(x\right)=\frac{2}{5}x-4\\\\=\frac{5}{2}\left(\frac{2}{5}x-4\right)+10\\\\=x[/tex]
so, both are inverses of each other.
c) Draw the graphs of f(x) and g(x) on the same coordinate plane.
As shown below in the graph , Since for inverse function we need an axis of symmetry i.e. y=x
And both f(x) and g(x) are symmetry to y=x.
∴ f(x) and g(x) are inverses of each other.