Problem 14 - College Algebra

Question

Use each pair of functions to find \(f(g(x))\) and \(g(f(x))\). Simplify your answers. $$ f(x)=|x|, g(x)=5 x+1 $$

Step-by-Step Solution

Answer

f(g(x)) = |5x + 1|, g(f(x)) = 5|x| + 1.

Step 1: Determine f(g(x))

First, identify the expression for the composition of the function \( f(x) \) with function \( g(x) \). This requires substituting \( g(x) \) into \( f(x) \). Given that \( f(x) = |x| \) and \( g(x) = 5x+1 \), the expression becomes \( f(g(x)) = |g(x)| = |5x + 1| \).

Step 2: Determine g(f(x))

Next, compute the composition of the function \( g(x) \) with \( f(x) \). Substitute \( f(x) \) into \( g(x) \), which becomes \( g(f(x)) = 5|x| + 1 \), as \( f(x) = |x| \).

Step 3: Simplify the Solutions

For both expressions, simplify them if possible. Here, \( f(g(x)) = |5x + 1| \) remains as is, and \( g(f(x)) = 5|x| + 1 \) cannot be simplified further since it already represents an elementary operation.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in mathematics. It is often represented as \( f(x) = |x| \), where \( x \) can be any real number. The absolute value of \( x \), denoted by \( |x| \), represents the distance of \( x \) from zero on the number line. This distance is always positive or zero, never negative.

• The absolute value of a positive number is the number itself.
• The absolute value of zero is zero.
• The absolute value of a negative number is its opposite, which is positive.

For example, \( |3| = 3 \) and \( |-3| = 3 \). When applying the absolute value function to another expression, such as \( |5x + 1| \), it transforms the entire expression within the bars into a non-negative value. This means that regardless of the sign of \( 5x + 1 \), the outer expression must be positive. This transformation is what makes the absolute value function unique and useful in modeling real-world scenarios where only non-negative values are meaningful.

Composition of Functions

The composition of functions is a process where one function is applied to the result of another function. The notation \( f(g(x)) \) denotes the composition of the function \( f \) with \( g \). This effectively means you take the output of \( g(x) \) and use it as the input for \( f(x) \).

• To find \( f(g(x)) \), substitute the entire \( g(x) \) expression into \( f(x) \).
• Conversely, \( g(f(x)) \) is found by substituting \( f(x) \) into \( g(x) \).

This way of combining functions allows us to create more complex functions and understand their behaviors. In the case of the exercise, by substituting \( g(x) = 5x + 1 \) into \( f(x) = |x| \), we form the expression \( f(g(x)) = |5x + 1| \). Similarly, \( g(f(x)) = 5|x| + 1 \). These new functions can reveal insights about the transformations applied to \( x \) and illustrate how different functions interact with each other.

Linear Functions

Linear functions are one of the simplest types of functions, having a constant rate of change and forming a straight line when graphed. A general linear function can be represented as \( g(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

• The slope \( m \) determines the steepness and direction of the line.
• The y-intercept \( b \) is where the line crosses the y-axis.

In the original exercise, \( g(x) = 5x + 1 \) is a linear function with a slope of 5 and a y-intercept of 1. This means that for each increase of 1 in \( x \), the value of the function increases by 5, and it crosses the y-axis at \( y = 1 \). Linear functions like this one are straightforward yet powerful tools in both mathematics and real-world applications. They can be used to model relationships with constant rates of change, such as velocity in physics or cost analysis in economics. By understanding their properties, we gain insights into how systems behave over time.