Question

Determine whether f(x)=5x+1/x and g(x)= x/5x+1 are inverse functions. Explain how you know.

Answer 1

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Answer 2

For f(x) and g(x) to be inverse functions, then $f(g(x)) = x$

f(x) and g(x) are not inverse functions

The functions are given as:

$f(x) = \frac{5x + 1}{x}$

$g(x) = \frac x{5x + 1}$

We have:

$f(x) = \frac{5x + 1}{x}$

Substitute g(x) for x

$f(g(x)) = \frac{5g(x) + 1}{g(x)}$

Substitute $g(x) = \frac x{5x + 1}$

$f(g(x)) = \frac{5 \times \frac x{5x + 1} + 1}{\frac x{5x + 1}}$

$f(g(x)) = \frac{\frac{5x}{5x + 1} + 1}{\frac x{5x + 1}}$

Take LCM

$f(g(x)) = \frac{\frac{5x + 5x + 1}{5x + 1}}{\frac x{5x + 1}}$

Cancel out 5x + 1

$f(g(x)) = \frac{5x + 5x + 1}{x}$

$f(g(x)) = \frac{10x + 1}{x}$

Recall that, the condition is:

$f(g(x)) = x$

This means that: f(x) and g(x) are not inverse functions

Read more about inverse functions at:

https://brainly.com/question/10300045