Math / Algebra

QuestionGiven the function $f(x)=x^{2}-16, x \geq 0$ , complete parts a through $c$ . (a) Find an equation for $f^{-1}(x)$ . (b) Graph $f$ and $f^{-1}$ in the same rectangular coordinate system. (c) Use interval notation to give the domain and the range of $f$ and $f^{-1}$ . (a) Find $\mathrm{f}^{-1}(\mathrm{x})$ . $f^{-1}(x)=\sqrt{x+16}$ (Type an exact answer, using radicals as needed.) (b) Graph $f$ and $\mathrm{f}^{-1}$ in the same coordinate system. Choose the correct graph below. A. B. C.

Studdy Solution

STEP 1

1. The function $f(x) = x^2 - 16$ is defined for $x \geq 0$ .
2. We need to find the inverse function $f^{-1}(x)$ .
3. We will graph both $f(x)$ and $f^{-1}(x)$ .
4. We will determine the domain and range of both $f(x)$ and $f^{-1}(x)$ .

STEP 2

1. Find the inverse function $f^{-1}(x)$ .
2. Graph $f(x)$ and $f^{-1}(x)$ .
3. Determine the domain and range of $f(x)$ and $f^{-1}(x)$ .

STEP 3

To find the inverse function $f^{-1}(x)$ , start by setting $y = f(x) = x^2 - 16$ .

STEP 4

Solve for $x$ in terms of $y$ : $y = x^2 - 16$ $y + 16 = x^2$ $x = \sqrt{y + 16}$ Since $x \geq 0$ , the inverse function is $f^{-1}(x) = \sqrt{x + 16}$ .

STEP 5

Graph the function $f(x) = x^2 - 16$ for $x \geq 0$ . This is a parabola opening upwards, starting at the point (0, -16).

STEP 6

Graph the inverse function $f^{-1}(x) = \sqrt{x + 16}$ . This is a square root function starting at the point (-16, 0) and increasing.

STEP 7

Determine the domain and range of $f(x)$ : - Domain of $f(x)$ : $[0, \infty)$ - Range of $f(x)$ : $[-16, \infty)$

STEP 8

Determine the domain and range of $f^{-1}(x)$ : - Domain of $f^{-1}(x)$ : $[-16, \infty)$ - Range of $f^{-1}(x)$ : $[0, \infty)$
The inverse function is: $f^{-1}(x) = \sqrt{x + 16}$
The domain and range are: - Domain of $f(x)$ : $[0, \infty)$ - Range of $f(x)$ : $[-16, \infty)$ - Domain of $f^{-1}(x)$ : $[-16, \infty)$ - Range of $f^{-1}(x)$ : $[0, \infty)$

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Studdy Solution

STEP 1

1. The function $f(x) = x^2 - 16$ is defined for $x \geq 0$ .
2. We need to find the inverse function $f^{-1}(x)$ .
3. We will graph both $f(x)$ and $f^{-1}(x)$ .
4. We will determine the domain and range of both $f(x)$ and $f^{-1}(x)$ .

STEP 2

1. Find the inverse function $f^{-1}(x)$ .
2. Graph $f(x)$ and $f^{-1}(x)$ .
3. Determine the domain and range of $f(x)$ and $f^{-1}(x)$ .

STEP 3

To find the inverse function $f^{-1}(x)$ , start by setting $y = f(x) = x^2 - 16$ .

STEP 4

Solve for $x$ in terms of $y$ : $y = x^2 - 16$ $y + 16 = x^2$ $x = \sqrt{y + 16}$ Since $x \geq 0$ , the inverse function is $f^{-1}(x) = \sqrt{x + 16}$ .

STEP 5

Graph the function $f(x) = x^2 - 16$ for $x \geq 0$ . This is a parabola opening upwards, starting at the point (0, -16).

STEP 6

Graph the inverse function $f^{-1}(x) = \sqrt{x + 16}$ . This is a square root function starting at the point (-16, 0) and increasing.

STEP 7

Determine the domain and range of $f(x)$ : - Domain of $f(x)$ : $[0, \infty)$ - Range of $f(x)$ : $[-16, \infty)$

STEP 8

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Studdy Solution

STEP 1

STEP 2

1. Find the inverse function f−1(x) f^{-1}(x) f−1(x).2. Graph f(x) f(x) f(x) and f−1(x) f^{-1}(x) f−1(x).3. Determine the domain and range of f(x) f(x) f(x) and f−1(x) f^{-1}(x) f−1(x).

STEP 3

To find the inverse function f−1(x) f^{-1}(x) f−1(x), start by setting y=f(x)=x2−16 y = f(x) = x^2 - 16 y=f(x)=x2−16.

STEP 4

Solve for x x x in terms of y y y: y=x2−16 y = x^2 - 16 y=x2−16 y+16=x2 y + 16 = x^2 y+16=x2 x=y+16 x = \sqrt{y + 16} x=y+16​ Since x≥0 x \geq 0 x≥0, the inverse function is f−1(x)=x+16 f^{-1}(x) = \sqrt{x + 16} f−1(x)=x+16​.

STEP 5

Graph the function f(x)=x2−16 f(x) = x^2 - 16 f(x)=x2−16 for x≥0 x \geq 0 x≥0. This is a parabola opening upwards, starting at the point (0, -16).

STEP 6

Graph the inverse function f−1(x)=x+16 f^{-1}(x) = \sqrt{x + 16} f−1(x)=x+16​. This is a square root function starting at the point (-16, 0) and increasing.

STEP 7

Determine the domain and range of f(x) f(x) f(x): - Domain of f(x) f(x) f(x): [0,∞) [0, \infty) [0,∞) - Range of f(x) f(x) f(x): [−16,∞) [-16, \infty) [−16,∞)

STEP 8

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1. Find the inverse function $f^{-1}(x)$ .
2. Graph $f(x)$ and $f^{-1}(x)$ .
3. Determine the domain and range of $f(x)$ and $f^{-1}(x)$ .

To find the inverse function $f^{-1}(x)$ , start by setting $y = f(x) = x^2 - 16$ .

Solve for $x$ in terms of $y$ : $y = x^2 - 16$ $y + 16 = x^2$ $x = \sqrt{y + 16}$ Since $x \geq 0$ , the inverse function is $f^{-1}(x) = \sqrt{x + 16}$ .

Graph the function $f(x) = x^2 - 16$ for $x \geq 0$ . This is a parabola opening upwards, starting at the point (0, -16).

Graph the inverse function $f^{-1}(x) = \sqrt{x + 16}$ . This is a square root function starting at the point (-16, 0) and increasing.

Determine the domain and range of $f(x)$ : - Domain of $f(x)$ : $[0, \infty)$ - Range of $f(x)$ : $[-16, \infty)$