Problem 86 - Precalculus : Building Concepts and Connections

Question

Let \(f(x)=2 x+5\) and \(g(x)=f(x+2)-4 .\) Graph both functions on the same set of coordinate axes. Describe the transformation from \(f(x)\) to \(g(x) .\) What do you observe?

Step-by-Step Solution

Answer

The transformation from \(f(x)\) to \(g(x)\) is a mathematical rearrangement, where inside the function \(f(x)\) , \(x\) is replaced by \(x + 2\) and subtracted by 4, giving \(g(x)\). However, this does not change the graph. Hence, graphically, the function \(f(x)\) is not transformed to obtain \(g(x)\), as after simplifying, \(f(x)\) and \(g(x)\) are the same. The graph is a straight line with gradient 2 and y-intercept 5.

Step 1: Define the Functions

Firstly, define the functions based on the problem statement. Let \(f(x)=2x+5\) and \(g(x)=f(x+2)-4\).

Step 2: Simplify Function g(x)

Substitute \(f(x)\) into the definition of \(g(x)\) to simplify it. Therefore, \(g(x) = f(x+2) - 4 = [2*(x+2)+5] - 4 = 2x + 4 + 5 - 4 = 2x + 5\). Hence, \(g(x) = 2x + 5\), which is the same as \(f(x)\).

Step 3: Graph f(x) and g(x)

Graph the functions \(f(x)\) and \(g(x)\) on the same set of coordinate axes. Since both the functions are same, you will get an identical line for both the functions.

Step 4: Describe the Transformation

The transformation from \(f(x)\) to \(g(x)\) appears to be a shuffling of terms rather than any change in the function itself. Thus, instead of any geometric transformation (like shifts or stretch), here it's present a mathematical transformation in terms of simplifying the expression. However, since the functions are the same (based on our previous simplification step), it means there's no actual transformation in terms from the graph of \(f(x)\) to \(g(x)\).

Key Concepts

Graphing Functions

Graphing functions is an essential skill in mathematics that allows us to visually understand and analyze the behavior of functions. Each function can be interpreted as a plot on a coordinate grid, providing insights into its characteristics, such as slope, intercepts, and any transformations applied. In our exercise, we are dealing with two functions: \(f(x) = 2x + 5\) and \(g(x) = f(x+2) - 4\). Graphing these functions involves plotting points for different values of \(x\) and illustrating them as a line on the grid.

Key elements to consider when graphing functions include:

• Slope: For linear functions, the slope determines the steepness and direction of the line. Here, both lines have the slope of 2.
• Y-intercept: The point where the line crosses the y-axis. In this example, both functions cross the y-axis at \(y = 5\).
• Transformation Insights: Examining transformations like shifts or reflections through graph comparison may reveal if any changes occur.

By graphing \(f(x)\) and \(g(x)\) on the same axis, it's easier to spot any transformations, although in this specific instance, these manipulations showed that \(f(x)\) and \(g(x)\) graph to the exact same line.

Linear Functions

Linear functions are a fundamental type of function in algebra, characterized by their constant slope and straight line graph. The standard form of a linear function is \(y = mx + b\), where \(m\) represents the slope, and \(b\) represents the y-intercept.

In our problem, both \(f(x) = 2x + 5\) and \(g(x) = 2x + 5\) are linear functions. They show the typical properties:

• Slope: The slope of 2 indicates that for every unit increase in \(x\), the function values rise by 2 units.
• Y-intercept: The y-intercept is 5, suggesting that when \(x = 0\), the function equals 5.

Understanding the components of linear functions aids in quickly determining their behavior and appearance on a graph. Even with transformations applied initially to \(f(x)\) to derive \(g(x)\), the simplification process proved they remain identical.

Function Simplification

Function simplification is a useful tool for altering complex function expressions into their simplest forms. This process often involves substituting values, performing arithmetic operations, and applying algebraic properties to streamline the expressions.

In the case of this exercise, the function \(g(x) = f(x+2)-4\) was simplified by substituting and expanding:

• Start with \(g(x) = f(x+2) - 4 = [2(x+2) + 5] - 4\).
• Distribute the 2 within the brackets: \(2(x+2) = 2x + 4\).
• Add the constants: \(2x + 4 + 5 = 2x + 9\).
• Subtract the outside constant: \(2x + 9 - 4 = 2x + 5\).

This simplification process confirmed that \(g(x) = 2x + 5\), which is identical to \(f(x)\). Through simplification, it becomes evident that transformations initially indicated through function notation can sometimes result in no change, as shown in this exercise, where the functions simplified to the exact form.