Math / Algebra

Question(A) Find a formula for the inverse of the function $f^{[-1]}(Q)=\square \quad f(x)=7-9 / x^{2}, \quad x>0$

Studdy Solution

STEP 1

1. We are given the function $f(x) = 7 - \frac{9}{x^2}$ .
2. We need to find the inverse function, denoted as $f^{[-1]}(Q)$ .
3. The domain of the function is $x > 0$ .

STEP 2

1. Set the function equal to $Q$ and solve for $x$ .
2. Rearrange the equation to express $x$ in terms of $Q$ .
3. Verify the domain and range of the inverse function.
4. Write the formula for the inverse function.

STEP 3

Set the function equal to $Q$ :
$Q = 7 - \frac{9}{x^2}$

STEP 4

Rearrange the equation to solve for $x^2$ :
$Q - 7 = -\frac{9}{x^2}$
Multiply both sides by $-1$ :
$7 - Q = \frac{9}{x^2}$

STEP 5

Solve for $x^2$ by multiplying both sides by $x^2$ and dividing by $7 - Q$ :
$x^2 = \frac{9}{7 - Q}$

STEP 6

Take the square root of both sides to solve for $x$ . Since $x > 0$ , we take the positive root:
$x = \sqrt{\frac{9}{7 - Q}}$

STEP 7

Verify the domain and range of the inverse function. Since $x > 0$ , $7 - Q > 0$ , which implies $Q < 7$ .

STEP 8

Write the formula for the inverse function:
$f^{[-1]}(Q) = \sqrt{\frac{9}{7 - Q}}$
The formula for the inverse function is:
$f^{[-1]}(Q) = \sqrt{\frac{9}{7 - Q}}$

Was this helpful?

Question(A) Find a formula for the inverse of the function $f^{[-1]}(Q)=\square \quad f(x)=7-9 / x^{2}, \quad x>0$

Studdy Solution

STEP 1

1. We are given the function $f(x) = 7 - \frac{9}{x^2}$ .
2. We need to find the inverse function, denoted as $f^{[-1]}(Q)$ .
3. The domain of the function is $x > 0$ .

STEP 2

1. Set the function equal to $Q$ and solve for $x$ .
2. Rearrange the equation to express $x$ in terms of $Q$ .
3. Verify the domain and range of the inverse function.
4. Write the formula for the inverse function.

STEP 3

Set the function equal to $Q$ :
$Q = 7 - \frac{9}{x^2}$

STEP 4

Rearrange the equation to solve for $x^2$ :
$Q - 7 = -\frac{9}{x^2}$
Multiply both sides by $-1$ :
$7 - Q = \frac{9}{x^2}$

STEP 5

Solve for $x^2$ by multiplying both sides by $x^2$ and dividing by $7 - Q$ :
$x^2 = \frac{9}{7 - Q}$

STEP 6

Take the square root of both sides to solve for $x$ . Since $x > 0$ , we take the positive root:
$x = \sqrt{\frac{9}{7 - Q}}$

STEP 7

Verify the domain and range of the inverse function. Since $x > 0$ , $7 - Q > 0$ , which implies $Q < 7$ .

STEP 8

Write the formula for the inverse function:
$f^{[-1]}(Q) = \sqrt{\frac{9}{7 - Q}}$
The formula for the inverse function is:
$f^{[-1]}(Q) = \sqrt{\frac{9}{7 - Q}}$

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.

Question(A) Find a formula for the inverse of the function f[−1](Q)=□f(x)=7−9/x2,x>0f^{[-1]}(Q)=\square \quad f(x)=7-9 / x^{2}, \quad x>0f[−1](Q)=□f(x)=7−9/x2,x>0

Studdy Solution

STEP 1

1. We are given the function f(x)=7−9x2 f(x) = 7 - \frac{9}{x^2} f(x)=7−x29​.2. We need to find the inverse function, denoted as f[−1](Q) f^{[-1]}(Q) f[−1](Q).3. The domain of the function is x>0 x > 0 x>0.

STEP 2

1. Set the function equal to Q Q Q and solve for x x x.2. Rearrange the equation to express x x x in terms of Q Q Q.3. Verify the domain and range of the inverse function.4. Write the formula for the inverse function.

STEP 3

Set the function equal to Q Q Q:Q=7−9x2 Q = 7 - \frac{9}{x^2} Q=7−x29​

STEP 4

Rearrange the equation to solve for x2 x^2 x2:Q−7=−9x2 Q - 7 = -\frac{9}{x^2} Q−7=−x29​Multiply both sides by −1-1−1:7−Q=9x2 7 - Q = \frac{9}{x^2} 7−Q=x29​

STEP 5

Solve for x2 x^2 x2 by multiplying both sides by x2 x^2 x2 and dividing by 7−Q 7 - Q 7−Q:x2=97−Q x^2 = \frac{9}{7 - Q} x2=7−Q9​

STEP 6

Take the square root of both sides to solve for x x x. Since x>0 x > 0 x>0, we take the positive root:x=97−Q x = \sqrt{\frac{9}{7 - Q}} x=7−Q9​​

STEP 7

Verify the domain and range of the inverse function. Since x>0 x > 0 x>0, 7−Q>0 7 - Q > 0 7−Q>0, which implies Q<7 Q < 7 Q<7.

STEP 8

Write the formula for the inverse function:f[−1](Q)=97−Q f^{[-1]}(Q) = \sqrt{\frac{9}{7 - Q}} f[−1](Q)=7−Q9​​ The formula for the inverse function is:f[−1](Q)=97−Q f^{[-1]}(Q) = \sqrt{\frac{9}{7 - Q}} f[−1](Q)=7−Q9​​

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.

Question(A) Find a formula for the inverse of the function $f^{[-1]}(Q)=\square \quad f(x)=7-9 / x^{2}, \quad x>0$

1. We are given the function $f(x) = 7 - \frac{9}{x^2}$ .
2. We need to find the inverse function, denoted as $f^{[-1]}(Q)$ .
3. The domain of the function is $x > 0$ .

1. Set the function equal to $Q$ and solve for $x$ .
2. Rearrange the equation to express $x$ in terms of $Q$ .
3. Verify the domain and range of the inverse function.
4. Write the formula for the inverse function.

Set the function equal to $Q$ :
$Q = 7 - \frac{9}{x^2}$

Rearrange the equation to solve for $x^2$ :
$Q - 7 = -\frac{9}{x^2}$
Multiply both sides by $-1$ :
$7 - Q = \frac{9}{x^2}$

Solve for $x^2$ by multiplying both sides by $x^2$ and dividing by $7 - Q$ :
$x^2 = \frac{9}{7 - Q}$

Take the square root of both sides to solve for $x$ . Since $x > 0$ , we take the positive root:
$x = \sqrt{\frac{9}{7 - Q}}$

Verify the domain and range of the inverse function. Since $x > 0$ , $7 - Q > 0$ , which implies $Q < 7$ .

Write the formula for the inverse function:
$f^{[-1]}(Q) = \sqrt{\frac{9}{7 - Q}}$
The formula for the inverse function is:
$f^{[-1]}(Q) = \sqrt{\frac{9}{7 - Q}}$