Question
The function \( f(x)=15 x \) is one-to-one. a. Find an equation for \( f^{-1}(x) \), the inverse function. b. Verify that your equation is correct by showing that \( f\left(f^{-1}(x)\right)=x \) and \( f^{-1}(f(x))=x \).
Ask by Murphy Best. in the United States
Feb 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
a. The inverse function is \( f^{-1}(x) = \frac{x}{15} \).
b.
- \( f(f^{-1}(x)) = x \)
- \( f^{-1}(f(x)) = x \)
This confirms that \( f^{-1}(x) = \frac{x}{15} \) is the correct inverse of \( f(x) = 15x \).
Solution
a. To find the inverse function f⁻¹(x) for f(x) = 15x, start by writing y = 15x. Then solve for x in terms of y:
y = 15x
x = y/15
Now, swapping x and y gives:
f⁻¹(x) = x/15
b. To verify that this is indeed the inverse function, we check both compositions.
First, calculate f(f⁻¹(x)):
f(f⁻¹(x)) = f(x/15) = 15 · (x/15) = x
Next, calculate f⁻¹(f(x)):
f⁻¹(f(x)) = f⁻¹(15x) = (15x)/15 = x
Since both compositions return x, we have confirmed that f⁻¹(x) = x/15 is the correct inverse of f(x) = 15x.
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Mind Expander
To find the inverse function \( f^{-1}(x) \) for \( f(x) = 15x \), we start by replacing \( f(x) \) with \( y \): \[ y = 15x. \] Next, we solve for \( x \) in terms of \( y \): \[ x = \frac{y}{15}. \] Thus, the inverse function is: \[ f^{-1}(x) = \frac{x}{15}. \] Now, let’s verify that \( f\left(f^{-1}(x)\right) = x \): 1. Calculate \( f\left(f^{-1}(x)\right) \): \[ f\left(f^{-1}(x)\right) = f\left(\frac{x}{15}\right) = 15\left(\frac{x}{15}\right) = x. \] 2. Next, calculate \( f^{-1}(f(x)) \): \[ f^{-1}(f(x)) = f^{-1}(15x) = \frac{15x}{15} = x. \] Since both identities hold true, we have verified that the inverse function is correct.
