Problem 1 - A Graphical Approach to College Algebra

Question

Give the (a) \(x\) -intercept, (b) \(y\) -intercept, (c) domain, (d) range, and (e) slope of the line. Do not use a calculator. $$f(x)=x-4$$

Step-by-Step Solution

Answer

(a) x-intercept: (4, 0), (b) y-intercept: (0, -4), (c) domain: all real numbers, (d) range: all real numbers, (e) slope: 1.

Step 1: Find the x-intercept

To find the x-intercept, we set the function \( f(x) = 0 \). Thus, the equation becomes \( x - 4 = 0 \). Solving for \( x \), we get \( x = 4 \). Hence, the x-intercept is at the point \( (4, 0) \).

Step 2: Find the y-intercept

The y-intercept of the function is found by setting \( x = 0 \) in the function. Substituting \( x = 0 \) into \( f(x) = x - 4 \), we get \( f(0) = 0 - 4 = -4 \). Thus, the y-intercept is at the point \( (0, -4) \).

Step 3: Determine the Domain

The domain of a linear function like \( f(x) = x - 4 \) is all real numbers, because there are no restrictions on the values that \( x \) can take.

Step 4: Determine the Range

The range of \( f(x) = x - 4 \) is also all real numbers. This is because for any real number \( y \), there exists a corresponding \( x \) such that \( f(x) = y \).

Step 5: Find the Slope

The slope of the line in the equation \( f(x) = x - 4 \) is the coefficient of \( x \, \). In this case, the slope is 1, indicating that for every 1 unit increase in \( x \), \( f(x) \) increases by 1 unit as well.

Key Concepts

Intercepts

Intercepts are crucial points on the graph of a function where the graph crosses the axes. For a linear equation, like the one given in the exercise, these are typically divided into two categories: the \(x\)-intercept and the \(y\)-intercept.

• X-intercept: This is the point where the graph crosses the \(x\)-axis. To find it, set the function equal to zero and solve for \(x\). For the function \(f(x) = x - 4\), set \(f(x) = 0\). Solving \(x - 4 = 0\) gives \(x = 4\). So, the \(x\)-intercept is at the point \((4, 0)\).
• Y-intercept: This is the point where the graph crosses the \(y\)-axis. To find it, substitute \(x = 0\) into the function. Substituting into \(f(x) = x - 4\) gives \(f(0) = -4\). So, the \(y\)-intercept is at the point \((0, -4)\).

Intercepts offer important insights into the behavior of linear functions and are essential for graphing these equations effectively.

Domain and Range

Understanding the domain and range of a function is vital as it tells us about all possible input and output values. For linear functions, the rules are quite straightforward.

• Domain: The domain refers to all the possible \(x\)-values that a function can take. For linear equations like \(f(x) = x - 4\), there are no restrictions. Thus, the domain is all real numbers, represented in set notation as \((-\infty, \infty)\).
• Range: The range involves all possible \(y\)-values that a function can produce. With linear functions, every real number \(x\) results in a unique \(y\)-value, covering all real numbers as a result. Therefore, the range is also \((-\infty, \infty)\).

Having a comprehensive understanding of the domain and range allows students to grasp the extent and limitations of a function, adding a deeper layer to their comprehension of linear equations.

Slope of a Line

The slope of a line is a fundamental aspect of linear equations that describes how steep the line is and the direction it moves. It essentially quantifies the change in \(y\) over the change in \(x\) along the line. In the function \(f(x) = x - 4\), the slope can be identified from the coefficient of \(x\). Here, the slope is \(1\), indicating a steady rise. This tells us that for every 1 unit increase in \(x\), the value of \(f(x)\) or \(y\) increases by 1 unit.

• A slope of 1 signifies a perfect diagonal line ascending from left to right.
• A positive slope, like this one, indicates that as \(x\) increases, \(y\) also increases.

Understanding slope is essential for predicting the angle and direction of a line, allowing for accurate interpretations and graphing of linear functions.