The functions in Exercises 93–95 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(f^-1(x)) = x and f^-1(f(x)) = x. f(x) = 4x - 3

Verified step by step guidance

1. Start with the given function: \(f(x) = 4x - 3\). To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = 4x - 3\).
2. Next, swap the roles of \(x\) and \(y\) to find the inverse: replace \(y\) with \(x\) and \(x\) with \(y\), giving \(x = 4y - 3\).
3. Solve this new equation for \(y\) to express the inverse function: add 3 to both sides to get \(x + 3 = 4y\), then divide both sides by 4 to isolate \(y\): \(y = \frac{x + 3}{4}\).
4. Rewrite \(y\) as \(f^{-1}(x)\), so the inverse function is \(f^{-1}(x) = \frac{x + 3}{4}\).
5. To verify the inverse, compute \(f(f^{-1}(x))\) by substituting \(f^{-1}(x)\) into \(f\): \(f\left(f^{-1}(x)\right) = 4 \times \frac{x + 3}{4} - 3\). Simplify this expression and check if it equals \(x\). Then compute \(f^{-1}(f(x))\) by substituting \(f(x)\) into \(f^{-1}\): \(f^{-1}(4x - 3) = \frac{(4x - 3) + 3}{4}\). Simplify and check if it equals \(x\).

Key Concepts

One-to-One Functions

A function is one-to-one if each output corresponds to exactly one input, meaning it passes the horizontal line test. This property ensures the function has an inverse because no two different inputs produce the same output.

Inverse Functions

The inverse of a function reverses the roles of inputs and outputs, denoted as f⁻¹(x). To find the inverse, solve the equation y = f(x) for x in terms of y, then interchange x and y. The inverse 'undoes' the original function.

Verification of Inverse Functions

To verify that two functions are inverses, show that composing them in either order returns the input: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This confirms that each function reverses the effect of the other.