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) = x +3

Verified step by step guidance

1. Start with the given function: \(f(x) = x + 3\). To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = x + 3\).
2. Next, interchange the variables \(x\) and \(y\) to reflect the inverse relationship: \(x = y + 3\).
3. Now, solve this equation for \(y\) to express the inverse function: subtract 3 from both sides to get \(y = x - 3\).
4. Rewrite \(y\) as \(f^{-1}(x)\) to write the inverse function explicitly: \(f^{-1}(x) = x - 3\).
5. To verify the inverse, compute \(f(f^{-1}(x))\) by substituting \(f^{-1}(x)\) into \(f\): \(f(f^{-1}(x)) = (x - 3) + 3 = x\). Then compute \(f^{-1}(f(x))\) by substituting \(f(x)\) into \(f^{-1}\): \(f^{-1}(f(x)) = (x + 3) - 3 = x\). Both compositions return \(x\), confirming the inverse is correct.

Key Concepts

One-to-One Functions

A one-to-one function assigns each input a unique output, ensuring no two different inputs share the same output. This property is essential for a function to have an inverse, as the inverse must reverse the mapping without ambiguity.

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, effectively undoing the original operation.

Verification of Inverse Functions

To confirm that two functions are inverses, you compose them in both orders: f(f⁻¹(x)) and f⁻¹(f(x)). If both compositions simplify to x, the identity function, the functions are true inverses, validating the correctness of the inverse equation.