Question

Consider a linear function f(x) = mx + b

Answer 1

Answers:

$f^{-1}(x) = \frac{x-b}{m}$

and

$h^{-1}(x) = \frac{x+9}{2}$

See note below

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Explanation:

First we replace f(x) with y. Then we swap x and y. Afterward, solve for y to determine the inverse function. The m value cannot be zero to avoid dividing by zero.

$f(x) = mx+b\\\\y = mx+b\\\\x = my+b\\\\x-b = my\\\\y = \frac{x-b}{m}\\\\f^{-1}(x) = \frac{x-b}{m}$

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For the formula h(x) = 2x-9, we see that m = 2 and b = -9

So the inverse for h(x) would be:

$h^{-1}(x) = \frac{x-b}{m}\\\\h^{-1}(x) = \frac{x-(-9)}{2}\\\\h^{-1}(x) = \frac{x+9}{2}$

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Note: The function $h^{-1}(x) = \frac{x+9}{2}$ is the same as saying $h^{-1}(x) = \frac{1}{2}x+\frac{9}{2}$ or $h^{-1}(x) = 0.5x+4.5$, so it will depend on what format your teacher wants.