The functions in Exercises 11-28 are all one-to-one. For each function,a. Find an equation for f^-1(x), the inverse function.b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = 2x + 3

Verified step by step guidance

1. Step 1: To find the inverse of the function \( f(x) = 2x + 3 \), start by replacing \( f(x) \) with \( y \). So, \( y = 2x + 3 \).
2. Step 2: Swap \( x \) and \( y \) in the equation to find the inverse. This gives \( x = 2y + 3 \).
3. Step 3: Solve for \( y \) in terms of \( x \). Subtract 3 from both sides to get \( x - 3 = 2y \).
4. Step 4: Divide both sides by 2 to isolate \( y \). This results in \( y = \frac{x - 3}{2} \).
5. Step 5: Verify the inverse by checking \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). Substitute \( f^{-1}(x) = \frac{x - 3}{2} \) into \( f(x) \) and vice versa to confirm.

Key Concepts

One-to-One Functions

A one-to-one function is a type of function where each output is produced by exactly one input. This means that if f(a) = f(b), then a must equal b. One-to-one functions have unique inverses, which is essential for finding the inverse function f^-1(x). Understanding this property is crucial for solving the problem as it ensures that the inverse function exists.

Finding Inverse Functions

To find the inverse function f^-1(x), you typically start by replacing f(x) with y, then solve for x in terms of y. After isolating x, you swap x and y to express the inverse function. This process allows you to derive the equation for the inverse, which is necessary for part (a) of the question.

Verification of Inverse Functions

Verifying that two functions are inverses involves showing that f(f^-1(x)) = x and f^-1(f(x)) = x. This means that applying one function after the other returns the original input. This verification is crucial for part (b) of the question, as it confirms that the derived inverse function is correct and adheres to the properties of inverse functions.