Problem 65 - Calculus Single Variable

Question

In Exercises \(65-68\), find a function \(f\) such that \(g \circ f=h\). \(g(x)=x^{2}+2, h(x)=x^{2}-8 x+18\)

Step-by-Step Solution

Answer

The function \(f(x) = x - 4\).

Step 1: Understanding the Composition

We are given the functions \(g(x) = x^2 + 2\) and \(h(x) = x^2 - 8x + 18\). We need to find a function \(f\) such that \((g \circ f)(x) = h(x)\). This means plugging \(f(x)\) into \(g(x)\) should yield \(h(x)\).

Step 2: Setting up the Equation

We start by expressing the composition: \(g(f(x)) = (f(x))^2 + 2\). Set this equal to \(h(x)\): \[(f(x))^2 + 2 = x^2 - 8x + 18\].

Step 3: Simplifying the Equation

Rearrange the equation:\[(f(x))^2 = x^2 - 8x + 16\]. This can be rewritten as \[(f(x))^2 = (x - 4)^2\].

Step 4: Solving for f(x)

Since \((f(x))^2 = (x-4)^2\), the solutions for \(f(x)\) are \(f(x) = x - 4\) or \(f(x) = -(x - 4)\). However, because we generally take the principal branch for square roots in real functions within this context, we choose \(f(x) = x-4\).

Step 5: Verification

Verify the solution by substituting \(f(x)\) back into the composition:\[g(f(x)) = g(x - 4) = (x - 4)^2 + 2 = x^2 - 8x + 16 + 2 = x^2 - 8x + 18 = h(x)\].This confirms our solution is correct.

Key Concepts

Inverse Functions

Inverse functions are pairs of functions that "undo" each other. If you have a function \( f(x) \) and its inverse \( f^{-1}(x) \), then applying \( f \) followed by \( f^{-1} \) (or vice versa) should return you to your original input value. This is expressed as:

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

In our exercise, although we don't directly work with inverse functions, understanding inverses helps in grasping how composition affects outputs. Essentially, the goal of our task was similar: finding \( f(x) \) such that applying \( g \) results in \( h(x) \). Although it's not an inverse operation, it's helpful to keep in mind that composition is about combining steps, which is conceptually the opposite of the inverse function idea, where each step cancels the other. Knowing about inverse functions can help you understand the mutual transformations between sets. When establishing such relations, think of inverses as the paths that lead you back to an initial state, while direct operations state the forward-path construction.

Quadratic Functions

Quadratic functions involve terms where the variable is squared. The typical form of a quadratic function is \( ax^2 + bx + c \). In the context of our given exercise, both the functions \( g(x) = x^2 + 2 \) and \( h(x) = x^2 - 8x + 18 \) are quadratic. Quadratic equations can be reformulated by completing the square, which allows us to express them in a form like \((x - p)^2 + q\). For example, in the solution, we noted that \( (f(x))^2 = (x-4)^2 \), helping us hinge the transformation onto a specific form. Understanding quadratics is crucial because they describe parabolic shapes in graph representation. Important features of these graphs include:

• The vertex - the turning point of the parabola.
• The axis of symmetry - a vertical line through the vertex.

Parabolas can open upwards or downwards based on the sign of \( a \) in \( ax^2 \). This theoretical understanding will strengthen your ability to manipulate and comprehend real-world applications of quadratic expressions.

Function Verification

Function verification involves checking if the operations or compositions we performed yield the desired result. It is an important step to validate our solution. Simply put, it answers whether the function you've derived meets the condition specified in your problem. In the given solution exercise, we needed to check whether our derived \( f(x) = x - 4 \) satisfied the equation \( g(f(x)) = h(x) \). To do this, substitute \( f(x) \) back into the composition: 1. Replace \( f(x) \) with \( x-4 \) and evaluate \( g(x - 4) \). 2. Verify that the result on the left side matches \( h(x) \), which was achieved: \[ g(f(x)) = (x - 4)^2 + 2 = x^2 - 8x + 16 + 2 = x^2 - 8x + 18 = h(x) \] This ensures the step-by-step equation transformation stayed consistent with the original functions. Verification ensures the transformation is logical and valid, it solidifies understanding and confirms correct application of algebraic manipulations. Without verification, you may end up with an incorrect function that does not meet the problem's requirements.