Question

Find functions \(f\) and \(g\) so that \(f \circ g=H\). \(H(x)=|2 x+1|\)

Short answer

f(y) = |y| and g(x) = 2x + 1.

Step-by-step solution

Understand the Given Function

The function provided is H(x) = |2x + 1|. This is an absolute value function where the expression inside the absolute value is a linear function.

Identify a Suitable Inner Function

We need to find a function g(x) such that H(x) can be expressed as f(g(x)). Since H(x) = |2x + 1|, we can set g(x) = 2x + 1.

Identify the Outer Function

Next, we need an outer function f(y) that when applied to g(x) yields H(x). Since g(x) = 2x + 1, substituting g(x) in H(x) gives H(x) = |g(x)|. Therefore, f(y) = |y|.

Check the Composition

Finally, verify the composition f(g(x)): f(g(x)) = f(2x + 1) = |2x + 1|. This matches H(x), confirming that our choices for f and g are correct.

Key concepts

absolute value function

The absolute value function is a fundamental concept in mathematics. It takes any given number and returns its non-negative value. This is represented by the symbol \(|x|\), which means no matter if x is positive or negative, \(|x|\) will always be positive.
For example:

• If x = 3, then \(|3| = 3\).
• If x = -3, then \(|-3| = 3\).

The absolute value function is especially important because it helps to measure the distance between numbers on a number line without considering their direction. In the provided exercise, \(|2x + 1|\) uses the absolute value function to ensure the expression inside is non-negative.

linear function

A linear function is any function that can be graphically represented as a straight line. Its general form is \(y = mx + b\), where m is the slope (the change in y when x increases by 1) and b is the y-intercept (the point where the line crosses the y-axis).
Some characteristics of linear functions include:

• A constant rate of change
• A straight-line graph
• Simple relationships between variables

In the exercise, the inner function \(g(x) = 2x + 1\) is a linear function. Here, the slope m is 2, and the y-intercept b is 1. It transforms the input x by first multiplying by 2 and then adding 1.

inner and outer functions

In function composition, you deal with inner and outer functions. The composition \(f \circ g(x)\) means applying the inner function \(g(x)\) first, then applying the outer function \(f(y)\) on the result of \(g(x)\). This process can help break down complex expressions into simpler parts.
For example, if \(|H(x) = 2x + 1|\), we can break it down as follows:

• Choose the inner function as \(g(x) = 2x + 1\), which is a linear transformation
• Choose the outer function as \(f(y) = |y|\), which applies the absolute value

This means we first apply the linear function to x, and then take the absolute value of the result, confirming \(|2x + 1|\) as required. The order in which these functions are applied is crucial to getting the correct outcome.