Problem 9 - Precalculus Mathematics for Calculus

Question

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$f(x)=x-1$$

Step-by-Step Solution

Answer

The domain and range of \(f(x) = x - 1\) are both \((-\infty, \infty)\).

Step 1: Understanding the Function

The given function is a linear function, specifically, \(f(x) = x - 1\). A linear function of the form \(f(x) = mx + c\) is represented by a straight line with a slope \(m\) and a y-intercept \(c\). In this function, the slope \(m\) is 1 and the y-intercept \(c\) is -1.

Step 2: Using a Graphing Calculator

To draw the graph of the function \(f(x) = x - 1\), enter it into a graphing calculator. The graph will be a straight line that crosses the y-axis at -1 and has a slope of 1, meaning it rises one unit for every one unit it moves to the right.

Step 3: Finding the Domain from the Graph

The domain of a function is the set of all possible input values (x-values). For a linear function, such as \(f(x) = x - 1\), there are no restrictions on the x values. Therefore, the domain of this function is all real numbers, denoted as \((-\infty, \infty)\).

Step 4: Finding the Range from the Graph

The range of a function is the set of all possible output values (y-values). Similar to the domain, for a linear function like \(f(x) = x - 1\), the function can produce all real numbers as output. Thus, the range is all real numbers, denoted as \((-\infty, \infty)\).

Key Concepts

Graphing Calculator

A graphing calculator is an invaluable tool when it comes to visualizing mathematical functions. It enables you to plot functions like a linear function, which is simply a straight line described by an equation such as \(f(x) = x - 1\). Using a graphing calculator to plot this function involves inputting the equation into the calculator.

When you input \(f(x) = x - 1\), the calculator draws a straight line extending infinitely in both directions. The line’s slope of 1 means that for every step to the right on the x-axis, the line rises one unit on the y-axis. The y-intercept, which is -1 in this case, indicates the point where the line crosses the y-axis.

Plotting the function using a graphing calculator provides a clear visual representation that helps identify key features of the function, such as the domain and range.

Domain of a Function

The domain of a function refers to all the possible input values, or x-values, that you can use with the function. For linear functions like \(f(x) = x - 1\), there are no restrictions on x-values.

This means the function can handle any real number as an input. Therefore, for \(f(x) = x - 1\), the domain is all real numbers, represented in mathematical notation as

• \((-\infty, \infty)\).

Visualizing this on a graph, you would see the line extending endlessly in both left and right directions along the x-axis. This unbounded extent confirms the function's comprehensive domain, allowing for exploration of input across the entire number line.

Range of a Function

The range of a function is the set of all possible output values, or y-values, it can produce. For our linear function \(f(x) = x - 1\), every real number can be an output. The graph demonstrates that as x-values increase or decrease without limit, so do the y-values.

Just as the domain is limitless, the range in this case also encompasses all real numbers. In mathematical terms, the range is also expressed as:

• \((-\infty, \infty)\).

This infinite vertical extension on both sides of the graph illustrates that any real value can be a result of the function. The line’s continual rise and fall confirm that \(f(x) = x - 1\) isn’t bound by ceilings or floors, providing vast possibilities for y-values.