Math / Algebra

QuestionFind the inverse of the function $f(x)=\sqrt{x}+4$ . $f^{-1}(x)=(x+4)^{2}, x \geq 4$ $f^{-1}(x)=(x-4)^{2}, x \geq 0$ $f^{-1}(x)=(x-4)^{2}, x \geq 4$ $f^{-1}(x)=(x+4)^{2}, x \geq 0$ Clear my selection

Studdy Solution

STEP 1

1. The function $f(x) = \sqrt{x} + 4$ is defined for $x \geq 0$ because of the square root.
2. The inverse function $f^{-1}(x)$ will reverse the operations of $f(x)$ .
3. We need to solve for $x$ in terms of $y$ where $y = f(x)$ .

STEP 2

1. Set up the equation for the inverse.
2. Solve for $x$ in terms of $y$ .
3. Determine the domain of the inverse function.
4. Verify the inverse function.

STEP 3

Set up the equation for the inverse by letting $y = f(x)$ :
$y = \sqrt{x} + 4$

STEP 4

Solve for $x$ in terms of $y$ :
$y - 4 = \sqrt{x}$
Square both sides to eliminate the square root:
$(y - 4)^2 = x$
Thus, the inverse function is:
$f^{-1}(y) = (y - 4)^2$

STEP 5

Determine the domain of the inverse function. Since $f(x) = \sqrt{x} + 4$ is defined for $x \geq 0$ , the range of $f(x)$ is $y \geq 4$ . Therefore, the domain of the inverse function $f^{-1}(x)$ is $x \geq 4$ .

STEP 6

Verify the inverse function by checking that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ .
1. $f(f^{-1}(x)) = f((x - 4)^2) = \sqrt{(x - 4)^2} + 4 = x$ for $x \geq 4$ .
2. $f^{-1}(f(x)) = f^{-1}(\sqrt{x} + 4) = (\sqrt{x} + 4 - 4)^2 = x$ for $x \geq 0$ .
Both conditions are satisfied, confirming the inverse function is correct.
The correct inverse function is $f^{-1}(x) = (x - 4)^2$ with $x \geq 4$ .

Was this helpful?

QuestionFind the inverse of the function $f(x)=\sqrt{x}+4$ . $f^{-1}(x)=(x+4)^{2}, x \geq 4$ $f^{-1}(x)=(x-4)^{2}, x \geq 0$ $f^{-1}(x)=(x-4)^{2}, x \geq 4$ $f^{-1}(x)=(x+4)^{2}, x \geq 0$ Clear my selection

Studdy Solution

STEP 1

1. The function $f(x) = \sqrt{x} + 4$ is defined for $x \geq 0$ because of the square root.
2. The inverse function $f^{-1}(x)$ will reverse the operations of $f(x)$ .
3. We need to solve for $x$ in terms of $y$ where $y = f(x)$ .

STEP 2

1. Set up the equation for the inverse.
2. Solve for $x$ in terms of $y$ .
3. Determine the domain of the inverse function.
4. Verify the inverse function.

STEP 3

Set up the equation for the inverse by letting $y = f(x)$ :
$y = \sqrt{x} + 4$

STEP 4

Solve for $x$ in terms of $y$ :
$y - 4 = \sqrt{x}$
Square both sides to eliminate the square root:
$(y - 4)^2 = x$
Thus, the inverse function is:
$f^{-1}(y) = (y - 4)^2$

STEP 5

Determine the domain of the inverse function. Since $f(x) = \sqrt{x} + 4$ is defined for $x \geq 0$ , the range of $f(x)$ is $y \geq 4$ . Therefore, the domain of the inverse function $f^{-1}(x)$ is $x \geq 4$ .

STEP 6

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

1. The function f(x)=x+4 f(x) = \sqrt{x} + 4 f(x)=x​+4 is defined for x≥0 x \geq 0 x≥0 because of the square root.2. The inverse function f−1(x) f^{-1}(x) f−1(x) will reverse the operations of f(x) f(x) f(x).3. We need to solve for x x x in terms of y y y where y=f(x) y = f(x) y=f(x).

STEP 2

1. Set up the equation for the inverse.2. Solve for x x x in terms of y y y.3. Determine the domain of the inverse function.4. Verify the inverse function.

STEP 3

Set up the equation for the inverse by letting y=f(x) y = f(x) y=f(x):y=x+4 y = \sqrt{x} + 4 y=x​+4

STEP 4

Solve for x x x in terms of y y y:y−4=x y - 4 = \sqrt{x} y−4=x​Square both sides to eliminate the square root:(y−4)2=x (y - 4)^2 = x (y−4)2=xThus, the inverse function is:f−1(y)=(y−4)2 f^{-1}(y) = (y - 4)^2 f−1(y)=(y−4)2

STEP 5

Determine the domain of the inverse function. Since f(x)=x+4 f(x) = \sqrt{x} + 4 f(x)=x​+4 is defined for x≥0 x \geq 0 x≥0, the range of f(x) f(x) f(x) is y≥4 y \geq 4 y≥4. Therefore, the domain of the inverse function f−1(x) f^{-1}(x) f−1(x) is x≥4 x \geq 4 x≥4.

STEP 6

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The function $f(x) = \sqrt{x} + 4$ is defined for $x \geq 0$ because of the square root.
2. The inverse function $f^{-1}(x)$ will reverse the operations of $f(x)$ .
3. We need to solve for $x$ in terms of $y$ where $y = f(x)$ .

1. Set up the equation for the inverse.
2. Solve for $x$ in terms of $y$ .
3. Determine the domain of the inverse function.
4. Verify the inverse function.

Set up the equation for the inverse by letting $y = f(x)$ :
$y = \sqrt{x} + 4$

Solve for $x$ in terms of $y$ :
$y - 4 = \sqrt{x}$
Square both sides to eliminate the square root:
$(y - 4)^2 = x$
Thus, the inverse function is:
$f^{-1}(y) = (y - 4)^2$

Determine the domain of the inverse function. Since $f(x) = \sqrt{x} + 4$ is defined for $x \geq 0$ , the range of $f(x)$ is $y \geq 4$ . Therefore, the domain of the inverse function $f^{-1}(x)$ is $x \geq 4$ .