Problem 124 - College Algebra with Modeling and Visualization

Question

Show that if \(f(x)=a x+b\) and \(g(x)=c x+d,\) then \((g \circ f)(x)\) also represents a linear function. Find the slope of the graph of \((g \circ f)(x)\)

Step-by-Step Solution

Answer

The slope of \((g \circ f)(x)\) is \(ac\).

Step 1: Identify Linear Functions

We start by identifying the linear functions provided. The function \( f(x) = ax + b \) is clearly a linear function in the form of \( y = mx + b \). Similarly, \( g(x) = cx + d \) is a linear function. Each function has a constant slope: \( f(x) \) has a slope of \( a \) and \( g(x) \) has a slope of \( c \).

Step 2: Compute the Composition (g ⚬ f)(x)

The composition \((g \circ f)(x)\) means we substitute \(f(x)\) into \(g(x)\). Thus, \((g \circ f)(x) = g(f(x)) = g(ax + b)\).

Step 3: Substitute and Simplify

Substitute \(f(x) = ax + b\) into \(g(x) = cx + d\). Therefore, \(g(ax + b) = c(ax + b) + d\). Simplify this expression: \(g(ax + b) = c(ax) + cb + d = acx + cb + d\).

Step 4: Identify the Resulting Function

The expression \(g(ax + b) = acx + (cb + d)\) is in the linear form \(y = mx + b\), where \(ac\) is the slope and \(cb + d\) is the y-intercept.

Step 5: Determine the Slope of (g ⚬ f)(x)

From the expression derived, \( acx + (cb + d)\), we see that the slope, \(m\), of the linear function \((g \circ f)(x)\) is \( ac \). Thus, \( (g \circ f)(x) \) represents a linear function with a slope of \(ac\).

Key Concepts

Linear Functions

Linear functions are a fundamental type of function represented by the equation \( y = mx + b \). They form a straight line when graphed on a coordinate plane. In this formula, \( m \) stands for the slope of the line, and \( b \) is the y-intercept, which is where the line crosses the y-axis.
Linear functions are simple yet powerful tools in mathematics because they model a constant rate of change.
For example:

• The function \( f(x) = ax + b \) is a linear function with slope \( a \) and y-intercept \( b \)
• Similarly, \( g(x) = cx + d \) is another linear function with slope \( c \) and y-intercept \( d \)

Understanding linear functions is crucial for mastering more complex mathematical concepts, as they often serve as building blocks for more intricate equations.

Function Composition

Function composition is the process of combining two functions to make a new function. It's like putting one function inside of another. This is commonly written as \((g \circ f)(x)\), which means you take the output from \(f(x)\) and use it as the input for \(g(x)\).
To compose the functions \( f(x) = ax + b \) and \( g(x) = cx + d \), perform the following steps:

• Start by substituting \(f(x)\) into \(g(x)\): \(g(f(x)) = g(ax+b)\)
• Replace \(x\) in \(g(x)\) with \(ax + b\): \(c(ax+b) + d\)
• Distribute and simplify to find \( g(f(x)) = acx + cb + d \)

This results in a new function that is also a linear function. The beauty of function composition lies in its versatility, allowing complex processes to be broken down into manageable parts.

Slope of a Line

The slope of a line is a measure of its steepness and direction. In the equation \( y = mx + b \), the slope \( m \) defines how steep the line is. The slope tells us how much the y-value changes for a given change in the x-value.
There are a few key points to remember about slopes:

• A positive slope means the line is rising, while a negative slope indicates it's falling
• The slope is calculated as \( \frac{\text{change in y}}{\text{change in x}} \)
• Zero slope corresponds to a horizontal line

In the context of function composition, when two linear functions \( f(x) = ax + b \) and \( g(x) = cx + d \) are composed, the slope of the resulting linear function \((g \circ f)(x)\) is \( ac \).
This is because the composition affects the rate of change of each individual function, resulting in a scaled version of their original slopes.