$f: [0, \infty) \to [0, \infty)$, $f(x) = x^2$. Swap: $x = y^2$. Solve (positive root, since range of $f^{-1}$ is $[0, \infty)$): $y = \sqrt{x}$.

Question 1

What is worked example?

Accepted Answer

$f(x) = \sqrt{x}$, $g(x) = x - 4$.

Question 2

What is finding $f^{-1}$ algebraically?

Accepted Answer

1. Write $y = f(x)$. 2. Swap $x$ and $y$. 3. Solve for $y$. 4.

Question 3

What is graphical relationship?

Accepted Answer

The graph of $f^{-1}$ is the reflection of the graph of $f$ in the line $y = x$.

Question 4

What is worked examples?

Accepted Answer

Linear. $f(x) = 3x + 2$. Swap: $x = 3y + 2$. Solve: $y = (x - 2)/3$. So $f^{-1}(x) = (x - 2)/3$.

Question 5

What is linear?

Accepted Answer

$f(x) = 3x + 2$. Swap: $x = 3y + 2$. Solve: $y = (x - 2)/3$.

Question 6

What is quadratic with restriction?

Accepted Answer

$f: [0, \infty) 	o [0, \infty)$, $f(x) = x^2$. Swap: $x = y^2$. Solve (positive root, since range of $f^{-1}$ is $[0, \infty)$): $y = \sqrt{x}$.

Question 7

What is exponential?

Accepted Answer

$f(x) = 2^x$. The inverse is $f^{-1}(x) = \log_2(x)$ on $(0, \infty)$.

Question 8

What is confusing $f^{-1} $ with $1/f $?

Accepted Answer

Inverse function vs reciprocal. Different things.

Question 9

What is forgetting domain restriction?

Accepted Answer

Asking for the inverse of $x^2$ on $\mathbb{R}$ is not well-defined. Restrict to make $f$ one-to-one.

Question 10

What is wrong root sign on quadratic inverse?

Accepted Answer

Solving $y = x^2$ gives $x = \pm\sqrt{y}$; the correct sign depends on the restricted domain.

Question 11

What is composite order confusion?

Accepted Answer

$(f \circ g)(x) = f(g(x))$: apply $g$ first. Many students apply $f$ first by mistake.

Question 12

What is domain ignored in composite?

Accepted Answer

$f(g(x))$ requires $x$ in the domain of $g$ AND $g(x)$ in the domain of $f$.