Question

a) Given \(f(x)=4-x,\) graph the functions \(y=f(x)\) and \(y=\sqrt{f(x)}\) b) Compare the two functions and explain how their values are related. c) Identify the domain and range of each function, and explain any differences.

Short answer

The functions \(y = 4 - x\) and \(y = \sqrt{4 - x}\) have different ranges; the former is limited by the requirement that 4 - x must be non-negative.

Step-by-step solution

- Understand the function

Given the function \(f(x) = 4 - x\), observe that it's a linear function with a negative slope of -1 and a y-intercept at (0, 4).

- Graph \(y = f(x)\)

To graph \(y = f(x) = 4 - x\), plot the y-intercept at (0, 4) and use the slope to find another point, such as (4, 0). Draw a straight line through these points.

- Determine the function \(y = \sqrt{f(x)}\)

Given \(y = \sqrt{f(x)}\), we need to graph \(y = \sqrt{4 - x}\). Remember to consider the domain of the square root function where 4 - x must be non-negative.

- Graph \(y = \sqrt{f(x)}\)

To graph \(y = \sqrt{4 - x}\), calculate a few points: when \(x = 0\), \(y = \sqrt{4} = 2\); when \(x = 4\), \(y = \sqrt{0} = 0\). Plot these points and draw the curve between them. The function is only defined for \(x \leq 4\) (non-negative values under the square root).

- Compare the functions

Compare \(y = 4 - x\) and \(y = \sqrt{4 - x}\): \(y = \sqrt{4 - x}\) is the square root of \(y = 4 - x\), creating a curve that lies below the line.

- Determine domain and range

For \(y = 4 - x\), the domain is all real numbers \((-\infty, \infty)\) and the range is also all real numbers \((-\infty, 4]\). For \(y = \sqrt{4 - x}\), the domain is \(x \leq 4\) and the range is \([0, 2]\). The domain of \(y = \sqrt{4 - x}\) is limited to \(x \leq 4\) due to the square root, which requires non-negative values under the root.

Key concepts

Linear Functions

Linear functions are a fundamental concept in algebra. They are functions of the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In our exercise, the linear function given is \(f(x) = 4 - x\). Here, the slope \(m\) is -1, indicating a downward slant, and the y-intercept \(b\) is 4, which is where the line crosses the y-axis.
To graph linear functions, simply follow these steps:

• Identify the y-intercept and plot this point on the graph.
• Use the slope to determine the direction and steepness of the line. For every 1 unit you move to the right on the x-axis, move vertically by the slope value. In this case, move 1 unit down because the slope is -1.
• Draw a straight line through the points plotted to complete the graph.

Linear functions are easy to handle and visualize, as they form straight lines. This makes them very useful for modeling relationships where change happens at a constant rate.

Square Root Functions

Square root functions add an interesting twist to graphing by introducing a non-linearity. They are typically of the form \g(x) = \sqrt{x}\. In our scenario, we are looking at \(y = \sqrt{4 - x}\). These functions are only defined for values that keep the expression under the square root non-negative.
To effectively graph square root functions:

• First, identify the domain. For \(y = \sqrt{4 - x}\), \4 - x\ must be greater than or equal to 0, which means \(x \leq 4\).
• Second, calculate key points by substituting values for \(x\) within the domain and finding the corresponding \(y\). For instance, \(x = 0\) gives \(y = 2\) and \(x = 4\) results in \(y = 0\).
• Plot these points and draw a smooth curve through them. The curve will start high (e.g., \(y = 2\)) and decrease, always staying non-negative.

Square root functions typically curve downwards, getting flatter as \(x\) increases, creating a shape known as a square root curve. This exercise helps students understand how such transformations work and their impact on graphs.

Domain and Range

Understanding the domain and range is crucial for graphing any function.

• The domain of a function refers to all possible inputs (x-values) that the function can accept. For linear functions like \f(x) = 4 - x\ the domain is all real numbers because there's no restriction on what \x\ can be. For \y = \sqrt{4 - x}\, the domain is restricted to \x \leq 4\ since we can't take the square root of a negative number.
• The range is all possible outputs (y-values) that the function can produce. For the linear function \f(x) = 4 - x\, the range is also all real numbers. For the square root function \y = \sqrt{4 - x}\, the range is \0\ to \2\ because the square root function can only produce non-negative results, and it reaches a maximum of 2 when \(x = 0\).

Domain and range provide vital information about the behavior and limits of a function, which is particularly important when dealing with more complex formulas. In our given problem, recognizing the differences in the domain and range between the linear and square root functions helps us understand why their graphs look different.