Problem 63 - Calculus Single Variable

Question

Find a function \(g\) such that \(g \circ f=h\) \(f(x)=x^{2}-9, h(x)=2 x^{2}\)

Step-by-Step Solution

Answer

The function is \(g(y) = 2y + 18\).

Step 1: Identify the Functions

We are given two functions, \(f(x) = x^2 - 9\) and \(h(x) = 2x^2\). Our goal is to find a function \(g(x)\) such that \(g(f(x)) = h(x)\).

Step 2: Substitute f(x) Into g(x)

We substitute \(f(x)\) into \(g\), which gives the expression \(g(x^2 - 9)\). Since we know \(g(f(x)) = h(x) = 2x^2\), we can write: \(g(x^2 - 9) = 2x^2\).

Step 3: Find the Argument of g

The left side of the equation \(g(x^2 - 9)\) indicates that \(g\) operates on \(x^2 - 9\). This means \(x^2 - 9\) is the input for \(g(x)\). To match \(h(x)\), solve \(y - 9 = x^2\) for \(g(y) = 2x^2\).

Step 4: Express x in Terms of g's Input

We have \(x^2 = y + b\). Rearranging gives \(g(y) = 2(y + 9)\) since \(x^2 - 9 = y\). This implies \(x^2 = y + 9\).

Step 5: Simplify g(y)

Based on the relationship \(x^2 = y + 9\), substitute into \(h(x) = 2x^2\) to find \(g(y) = 2(y + 9)\). Thus, \(g(y) = 2y + 18\).

Key Concepts

Inverse Functions

Inverse functions are a fundamental concept in algebra. They essentially "undo" whatever operation a function performs. If you have a function, say \(f(x)\), which maps an input \(x\) to an output \(y\), then the inverse of this function, denoted as \(f^{-1}(x)\), will map \(y\) back to \(x\).
One of the most intuitive ways to understand inverse functions is to think of them as a mirror reflection over the line \(y = x\). This perspective helps us grasp that if \(f(a) = b\), then \(f^{-1}(b) = a\).
When dealing with inverse functions, ensure that the function you are finding the inverse for is one-to-one (bijective). A function must pass the horizontal line test to have an inverse. That means that no horizontal line should intersect the graph of the function more than once.
In the example given, we're not directly finding the inverse, but understanding that for any element in the domain, the output must match the composition\(h(x)\). This understanding is crucial when aligning it with the expected function, \(g(x)\).

Function Substitution

Function substitution involves replacing one function with another. It's a bit like replacing variables in an equation with a known expression or value. In our problem, we substitute \(f(x)\) with its expression, \(x^2 - 9\), in the function \(g\). This step-wise replacement is vital to create an equation that \(g(x)\) must satisfy.
The goal of using substitution is to make solving equations easier or to recast problems into a familiar form. In this case, the substitution helps us find the mystery function \(g(x)\) such that \(g(f(x)) = h(x)\).

• First, identify where substitutions occur. In our example, \(h(x)\) needs to be expressed in terms of \(f(x)\).
• Substitute \(f(x) = x^2 - 9\) into the function \(g\) and equate it to \(h(x)\). This helps us isolate the form of \(g(y)\).

By performing this substitution, the problem simplifies and allows us to eventually solve for \(g(y)\).

Quadratic Functions

Quadratic functions are polynomial functions of degree two. A typical quadratic function has the form \(ax^2 + bx + c\) where \(a, b,\) and \(c\) are constants, and \(a eq 0\). These functions are characterized by their U-shaped graphs called parabolas.
In our exercise, both \(f(x) = x^2 - 9\) and \(h(x) = 2x^2\) are quadratic. Recognizing these as quadratics is crucial as they each have distinct properties like vertex form and axis of symmetry that hint at how they behave.
Quadratic functions can be solved by:

• Factoring
• Using the quadratic formula
• Completing the square

In terms of our problem, understanding that both functions \(f\) and \(h\) are quadratic helps us identify that \(g(x)\) needs to handle transformations of one quadratic to another. Ultimately, the quadratic relationship determines the linear transformation present in the solution for \(g(y)\).