Problem 27 - Algebra 2

Question

Let \(g(x)=2 x\) and \(h(x)=x^{2}+4 .\) Evaluate each expression. $$ (g \circ h)(-1) $$

Step-by-Step Solution

Answer

So, \((g \circ h)(-1)=10\).

Step 1: Evaluate the Inner Function

Start by evaluating \(h(x)\) at \(x=-1\). \[h(-1)=(-1)^{2}+4=1+4=5\]

Step 2: Evaluate the Outer Function

Next, substitute the result \(h(-1)=5\) into the outer function \(g(x)\). \[(g \circ h)(-1)=g(h(-1))=g(5)=2 \cdot 5=10\]

Key Concepts

Evaluation of Functions

Function evaluation is the process of determining the output of a function for a specific input. Functions are like machines: you input a number, and the function processes it to give you an output. For example, with the function \( g(x) = 2x \), if we input \( x = 3 \), we would get \( g(3) = 2 \times 3 = 6 \). This tells us what the function's value is when \( x \) is 3.
In this exercise, we are given two functions, \( g(x) = 2x \) and \( h(x) = x^2 + 4 \). Evaluating \( h(x) \) at \( x = -1 \) is the first step to find the output of this function, essentially solving \( h(-1) = (-1)^2 + 4 \), which equals 5.

Inner Function

The inner function, in a function composition, is the function that is computed first in a sequence of operations. In our example, \( h(x) \) is the inner function. It is called "inner" because it's inside the parentheses of the composition, and it gets evaluated first.
To illustrate, when evaluating \((g \circ h)(-1)\), we start with \( h(-1) \). This means we plug \( -1 \) into \( h(x) = x^2 + 4 \), which results in \( h(-1) = 5 \).
Understanding which function is the inner one is crucial because it defines the first operation in the sequence of function compositions.

Outer Function

The outer function is calculated based on the result of the inner function. For the given functions, \( g(x) \) acts as the outer function in the composition \((g \circ h)\).
After evaluating the inner function \( h(x) \) and getting the output, we take this result and use it as the input for the outer function. So for \( (g \circ h)(-1) \), after finding \( h(-1) = 5 \), we then evaluate \( g(5) \).
This leads to computing \( g(5) = 2 \cdot 5 = 10 \), showing the final output after the composition has been fully evaluated.

Substitution

Substitution is a key step in function composition, where the output from the inner function becomes the input for the outer function. This process allows us to connect multiple functions in sequence.
In our example, after we find \( h(-1) = 5 \), we substitute this result into \( g(x) \). Instead of using \( x \), we replace it directly with the output of the inner function, writing \( g(5) \).
Substitution provides the bridge between the different functions in the composition, ensuring that their combined operations are seamless and coherent.