Question

What is the inverse?

Answer 1

as you already know, to get the inverse of any expression we start off by doing a quick switcheroo on the variables and then solving for "y", let's do so.

`\stackrel{f(x)}{y}\hspace{5em}\stackrel{\textit{quick switcheroo}}{x~~ = ~~√(4y+5)}\implies x^2=4y+5\implies x^2-5=4y \\\\\\ \cfrac{x^2-5}{4}=y\implies \cfrac{1}{4}(x^2-5)=\stackrel{f^(-1)(x)}{y}\qquad x\geqslant 0`

now, I don't see any good reason why x ⩾ 0 or x ⩽ 0 , since any negative or positive values are perfectly valid.

Answer 2

`\textsf{C)} \quad f^(-1)(x)=(1)/(4)(x^2-5), \;\; x \geq 0`

Explanation:

Given function:

`f(x)=√(4x+5)`

The domain of the given function is restricted to x ≥ -⁵/₄.

Therefore, the range of the given function is restricted to f(x) ≥ 0.

The inverse of a function is its reflection in the line y = x.

To find the inverse of a function, swap x and y:

`\implies x=√(4y+5)`

Rearrange the equation to make y the subject:

`\implies x^2=4y+5`

`\implies 4y=x^2-5`

`\implies y=(1)/(4)(x^2-5)`

Replace y with f⁻¹(x):

`\implies f^(-1)(x)=(1)/(4)(x^2-5)`

The domain of the inverse of a function is the range of the original function.

Therefore, the domain of the inverse function is x ≥ 0.

Therefore, the inverse of the given function is:

`f^(-1)(x)=(1)/(4)(x^2-5), \;\; x \geq 0`