Problem 19 - College Algebra

Question

Use each set of functions to find \(f(g(h(x)))\). Simplify your answers. \(f(x)=x^{2}+1, g(x)=\frac{1}{x},\) and \(h(x)=x+3\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{(x+3)^2} + 1\).

Step 1: Find h(x)

The first function we need to evaluate is \( h(x) = x + 3 \). This means we are going to replace the variable \( x \) with \( x + 3 \) in the next function.

Step 2: Compute g(h(x))

Next, substitute \( h(x) \) in \( g(x) \). Here \( g(x) = \frac{1}{x} \), so if \( h(x) = x + 3 \), we get \( g(h(x)) = \frac{1}{x+3} \).

Step 3: Evaluate f(g(h(x)))

Now we substitute \( g(h(x)) \) into \( f(x) \). The function \( f(x) = x^2 + 1 \) becomes \[ f\left(\frac{1}{x+3}\right) = \left(\frac{1}{x+3}\right)^2 + 1. \] Simplifying the expression results in \( \frac{1}{(x+3)^2} + 1 \).

Key Concepts

Function Composition

A composite function is made by combining two or more functions in such a way that the output of one function becomes the input of another. When you see a notation like \(f(g(x))\), it means first you apply \(g(x)\), and then \(f(x)\) on the result.
Function composition is like nesting operations; you perform one, get the result, and then use this result for the next function. In the exercise example, we go a step further by finding \(f(g(h(x)))\). Here are the steps broken down:

• Start with the innermost function, which is \(h(x)\). Compute \(h(x)\) first.
• Use the result of \(h(x)\) to compute \(g(h(x))\).
• Finally, take the output from \(g(h(x))\) and use it in \(f(x)\) to find \(f(g(h(x)))\).

This step-by-step substitution is crucial in understanding how composite functions work. You are essentially layering functions to get a final output.

Function Evaluation

Evaluating a function means finding the output of a function for a particular input. This input could be a simple variable like \(x\), or even another function, which is the essence of composite functions.
In the exercise solution, \( g(h(x)) = \frac{1}{x+3} \) is an example of function evaluation where the output of \( h(x) \) becomes the input for \( g(x) \). Similarly, when evaluating \( f\left(g(h(x))\right) \), you substitute \( g(h(x)) \)'s value into \( f(x) \).
Function evaluation requires careful arithmetic and substitution. Be mindful of parentheses and the correct order of operations. Missteps here can lead to entirely different outcomes. Practice by trying simple and then more complex compositions, until you get comfortable with the substitution method.

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of powers of variables with coefficients. These are usually in forms like \( ax^n + bx^{n-1} + \ldots + c \).
In our example, \( f(x) = x^2 + 1 \) is a polynomial function, specifically a quadratic function because the highest power of \(x\) is 2.
When using polynomial functions in function composition, always pay attention to how the input affects the polynomial's structure. For \( f(g(h(x))) \) in our exercise, the input, \( \frac{1}{x+3} \), transforms the polynomial into:

• Substitute \( \frac{1}{x+3} \) into \( f(x) = x^2 + 1 \)
• Resulting in \( \left(\frac{1}{x+3}\right)^2 + 1 \), which simplifies further

Polynomial functions add a layer of complexity due to their multi-term nature and the presence of powers, but following the rules of composition and evaluation makes handling them straightforward.