Question

Find the inverse of the function f(x) =x^2-5. Is the inverse a function?

Answer 1

The inverse of the function f(x) = x^2 - 5 is found by swapping x and y and solving for y, resulting in y = ±√(x + 5). To ensure that the inverse is also a function, one must restrict the domain of the original function to non-negative values of x. Therefore, the inverse function, with the domain restricted, is f^{-1}(x) = √(x + 5).

Step-by-step explanation:

To find the inverse of the function f(x) = x^2 - 5, we need to swap the x and y and solve for y. Let's start by replacing f(x) with y:

• y = x^2 - 5

Now we swap the variables:

• x = y^2 - 5

Next, we solve for y:

1. Add 5 to both sides: x + 5 = y^2
2. Take the square root of both sides: y = ±√(x + 5)

However, since we need the inverse function to also be a function, it can have only one output for each input. The expression y = ±√(x + 5) implies two possible outputs for each input (one positive and one negative). To ensure that the inverse is a function, we must restrict the domain of the original function. For the function f(x) = x^2 - 5, if we restrict the domain to x ≥ 0, then the inverse will be:

• y = √(x + 5)

Which is a function since for each x, there is only one possible y value. So, for f(x) = x^2 - 5 with x ≥ 0, the inverse function is f^{-1}(x) = √(x + 5), and it is a function.

Answer 2

y=x^2-5, solve for x

y+5=x^2

x=√(y+5) so

f^-1(x)=√(y+5)