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Varvara68 [4.7K]
3 years ago
7

Consider the function f(x)= 2/5x-4

Mathematics
1 answer:
Gre4nikov [31]3 years ago
5 0

Answer: a) \frac{5}{2}x+10=f^{-1}(x)=g(x)

Step-by-step explanation:

Since we have given that

f(x)=\frac{2}{5}x-4

a.) Find the inverse of f(x) and name it g(x).

Let f(x) = y

So, it becomes

y=\frac{2}{5}x-4

Switching x to y , we get

x=\frac{2}{5}y-4

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)

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

\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

Similarly,

\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

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.


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