Dickson Haynes
02/22/2024 · Middle School
Find the inverse of \( f(x)=x^{2}-5 \)
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Tutor-Verified Answer
Step-by-step Solution
To find the inverse of the function \( f(x) = x^2 - 5 \), we first need to express \( y \) in terms of \( x \):
\[
y = x^2 - 5
\]
Next, we solve for \( x \) in terms of \( y \):
1. Add 5 to both sides:
\[
y + 5 = x^2
\]
2. Take the square root of both sides. Since \( f(x) = x^2 - 5 \) is a quadratic function, it is not one-to-one over all real numbers. To find the inverse, we need to restrict the domain. We can restrict \( f(x) \) to \( x \geq 0 \) (the right half of the parabola) to ensure it is one-to-one:
\[
x = \sqrt{y + 5}
\]
Now we can express the inverse function. Since we restricted the domain to \( x \geq 0 \), the inverse function is:
\[
f^{-1}(y) = \sqrt{y + 5}
\]
To express the inverse function in terms of \( x \), we can replace \( y \) with \( x \):
\[
f^{-1}(x) = \sqrt{x + 5}
\]
Thus, the inverse of the function \( f(x) = x^2 - 5 \) (with the domain restricted to \( x \geq 0 \)) is:
\[
f^{-1}(x) = \sqrt{x + 5}
\]
Quick Answer
The inverse of \( f(x)=x^{2}-5 \) is \( f^{-1}(x) = \sqrt{x + 5} \) for \( x \geq 0 \).
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