Question

In Exercises \(5-8\), find the inverse function informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=x+7\)

Short answer

The inverse function is \(f^{-1}(x) = x-7\). Both properties, \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\), are verified.

Step-by-step solution

Find the inverse function

To find the inverse function, first replace \(f(x)\) with \(y\). This gives us \(y=x+7\). To find the inverse, we switch \(x\) and \(y\) and solve for \(y\). This gives us the equation \(x=y+7\), which when solved for \(y\) gives us \(y=x-7\). Hence, the inverse function is \(f^{-1}(x)=x-7\).

Verify the first property

To verify the property that \(f(f^{-1}(x)) = x\), substitute \(f^{-1}(x)\) into the function \(f(x)\). Substituting we get \(f(f^{-1}(x))=f(x-7)=(x-7)+7=x\), which verifies the first property.

Verify the second property

To verify the property that \(f^{-1}(f(x)) = x\), substitute \(f(x)\) into the inverse function \(f^{-1}(x)\). This gives \(f^{-1}(f(x)) = f^{-1}(x+7) = (x+7)-7 = x\), which verifies the second property.

Key concepts

Algebraic Operations

Algebraic operations refer to the basic computational techniques used in algebra, such as addition, subtraction, multiplication, division, and finding roots. These operations are vital when dealing with functions, as they allow for the transformation and manipulation of equations to solve for unknowns or to simplify expressions.

In the context of finding an inverse function, algebraic operations help us reverse the effect of the original function. For instance, considering the function from the textbook exercise, \( f(x) = x + 7 \), we apply subtraction (an algebraic operation) to both sides to isolate the variable \(x\) and effectively 'undo' the addition by 7. This results in the inverse function \( f^{-1}(x) = x - 7 \), where addition is countered by subtraction.

Function Composition

Function composition involves combining two or more functions such that the output of one function becomes the input for another. It is represented by the notation \( (f \circ g)(x) = f(g(x)) \) for two functions \(f\) and \(g\).

The exercise we’re looking at employs function composition to verify the correctness of the inverse function. For an original function \( f \) and its inverse \( f^{-1} \), composing them - that is, feeding the output of one into the input of the other - should yield the identity function. This means that applying both functions successively, you are essentially doing nothing to the input value: \( (f \circ f^{-1})(x) = f(f^{-1}(x)) = x \) and similarly, \( (f^{-1} \circ f)(x) = f^{-1}(f(x)) = x \).

Verifying Inverse Functions

Verifying inverse functions is critical to ensure that one function indeed reverses the action of another. Two main properties demonstrate this: \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). These properties confirm that for every input \(x\), the composition of a function and its inverse yields the original input.

In the given exercise, the verification is a two-step process. First, we show that the result of substituting the inverse function into the original function gives us back our input \(x\). Then, we demonstrate that putting the original function into the calculated inverse also results in the input \(x\). The step-by-step solution explicitly walks through these checks, and both properties holding true is crucial evidence that the inverse function is correctly determined.