State the defining relationship between a function \(f\) and its inverse \(f^{-1}\).

If \(f(3)=10\), what is \(f^{-1}(10)\), assuming the inverse exists?

What condition must a function satisfy to have an inverse function on its stated domain?

State the horizontal line test in words.

Find the inverse of \(f(x)=x+4\).

Find the inverse of \(f(x)=3x\).

Find the inverse of \(f(x)=2x-5\).

Verify that \(f(x)=x-7\) and \(g(x)=x+7\) are inverse functions.

Find the inverse of \(f(x)=\frac{x-1}{2}\).

Find the inverse of \(f(x)=\frac{2x+3}{5}\).

Explain why \(f(x)=x^2\) on \(\mathbb R\) does not have an inverse function.

Restrict \(f(x)=x^2\) to \(x\ge0\). Find its inverse.

Find the inverse of \(f(x)=\frac{1}{x-2}\), including restrictions.

Show algebraically that \(f(x)=5x-1\) and \(g(x)=\frac{x+1}{5}\) are inverses.

Find the inverse of \(f(x)=\frac{3x-4}{x+2}\), stating excluded values for the inverse.

A function has inverse \(f^{-1}(x)=2x+1\). Find \(f(x)\).

The position model \(s(t)=4t+1\) has domain \(t\ge0\). Find the inverse and interpret it.

Diagnose the error: to find the inverse of \(f(x)=x^2\), a student writes \(f^{-1}(x)=\pm\sqrt{x}\).

Prove that if \(f\) has an inverse function, then \(f\) is one-to-one.

Find a domain restriction for \(f(x)=(x-2)^2+3\) that makes it invertible, then find the inverse on that restricted domain.