Problem 51 - Intermediate Algebra

Question

Find \(f(x)\) and \(g(x)\) such that \(h(x)=(f \circ g)(x) .\) Answers may vary. See Example 5. $$ h(x)=x^{5}+9 $$

Step-by-Step Solution

Answer

One solution is \( f(x) = x^5 \) and \( g(x) = x + 9 \).

Step 1: Understanding Function Composition

The problem requires us to express the function \( h(x) = x^5 + 9 \) as a composition of two functions, \( h(x) = (f \circ g)(x) \), where \( (f \circ g)(x) = f(g(x)) \). This means that we need to find two functions \( f(x) \) and \( g(x) \) such that substituting \( g(x) \) into \( f(x) \) results in \( x^5 + 9 \).

Step 2: Choosing a Simpler Inner Function

Choose a simpler form for \( g(x) \) as it will be substituted into \( f(x) \). Let's set \( g(x) = x + 9 \), which is based on the constant term in \( h(x) \).

Step 3: Determining the Outer Function

Now, we need to determine \( f(x) \) such that \( f(g(x)) = h(x) \). Substitute \( g(x) = x + 9 \) into \( f \), resulting in \( f(x+9) = h(x) = (x+9)^5 \). This can be rewritten such that \( f(y) = y^5 \), where \( y = g(x) = x + 9 \). Thus, \( f(x) = x^5 \).

Step 4: Verify the Composition

Verify the composition by substituting \( g(x) \) into \( f(x) \): \( f(g(x)) = f(x+9) = (x+9)^5 \). This matches \( h(x) = x^5 + 9 \), confirming that our choices are correct.

Key Concepts

Inner Function

In function composition, the inner function is the function which is inserted first before another function is applied on it.
To solve for function composition, start by selecting a plausible and simple choice for the inner function. Then figure out how it interacts with another function (the outer function) to recreate the original problem.
In our example, we begin by recognizing that the inner function, denoted as \( g(x) \), will be a simpler version that can influence the form of \( (f \circ g)(x) \).
We choose \( g(x) = x + 9 \) because it cleverly handles the constant present in our main function \( h(x) = x^5 + 9 \).

• The addition of 9 shifts the input value before other transformations.
• This simplification helps set up the later step of formulating the outer function.

Outer Function

After establishing an inner function, the outer function molds the outcome into the desired final form.
The outer function, \( f(x) \), is designed to operate on the output from the inner function, \( g(x) \).
In our task, we need \( f(x) \) so that \( f(g(x)) = h(x) \). By substituting \( g(x) = x + 9 \) into this equation, we find that \( f(x+9) = h(x) = (x+9)^5 \).

• This leads to \( f(y) = y^5 \), where \( y = g(x) = x+9 \).
• Effectively, \( f(x) = x^5 \), since the power operation turns the input from the inner function into part of \( h(x) \).

In function composition, precise selection of the outer function is crucial as it 'sculpts' the nested input into the final structure we seek.

Verify Composition

Verification is an essential step in function composition to ensure that the combination of the inner and outer functions correctly reproduces the original function, \( h(x) \).
To confirm this, substitute \( g(x) \) into \( f(x) \):
\( f(g(x)) = f(x + 9) = (x + 9)^5 \). Since \( (x + 9)^5 \) matches with \( h(x) \), our function choices are valid.

• Verification protects against miscalculations and validates our function definitions.
• This step assures that the problem-solving process has been executed correctly.

Function composition heavily relies on confirming one’s result by mathematically proving the sequence is correctly constructed.

Function Notation

Function notation is a systematic way to denote and manipulate functions.
It uses symbols such as \( f(x) \), \( g(x) \), and \( h(x) \) to represent different functions respectively.
When discussing compositions, standard notation is \( (f \circ g)(x) = f(g(x)) \).

• This shows that 'g' is the inner function worked first, then 'f', the outer function, directs the overall form.
• Notation aids in providing clear instruction concerning function order and proceedings.

Use of such symbolism is vital in mathematics for cleanly organizing and executing compositional tasks, clearly presenting how components mesh together in wider calculations.