Question

This is algebra someone help me

Answer 1

Answer: No, they are not inverses of each other.

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Step-by-step explanation

To check the claim of one function being the inverse of the other, we need to do function composition. The goal is to determine these two facts:

• f(g(x)) = x
• g(f(x)) = x

Those two statements must be true if f(x) is the inverse of g(x), and vice versa.

Let's plug g(x) into f(x) like so:

`f(\text{x}) = \sqrt{\text{x}}+4\\\\f(g(\text{x})) = \sqrt{g(\text{x})}+4\\\\f(g(\text{x})) = \sqrt{\text{x}^2+5}+4\\\\`

This is as far as we can go.

We cannot simplify this to x, so this proves that the two functions are NOT inverses of each other.

If you wanted, graph that out to see a nonlinear curve, which is in sharp contrast to the straight line y = x. This is visual proof that
`\sqrt{\text{x}^2+5}+4` cannot possibly simplify to x.

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Here's further proof. We'll look at plugging f(x) into g(x)

`g(\text{x}) = \text{x}^2 + 5\\\\g(f(\text{x})) = (f(\text{x}))^2 + 5\\\\g(f(\text{x})) = (\sqrt{\text{x}}+4)^2 + 5\\\\g(f(\text{x})) = \text{x}+8\sqrt{\text{x}} + 5\\\\`

That doesn't lead to x either. Therefore, functions f and g are not inverses of each other.

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Here's another approach.

Let's say we plugged x = 0 into f(x).

`f(\text{x}) = \sqrt{\text{x}}+4\\\\f(0) = √(0)+4\\\\f(0) = 4\\\\`

That output is then plugged in as the input of g(x). If g(x) was the inverse of f(x), then the output of g(4) should be 0. This is because inverses are designed to undo each other.

`g(\text{x}) = \text{x}^2 + 5\\\\g(4) = 4^2 + 5\\\\g(4) = 21\\\\`

We do not arrive at 0, which is further proof the two functions are not inverses of each other. I'll let you try other values to see what happens.