Problem 43 - Precalculus Mathematics for Calculus

Question

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$s(x)=\frac{4-3 x}{x+7}$$

Step-by-Step Solution

Answer

Domain: \(x \neq -7\). Range: \(y \neq -3\). Intercepts: x-intercept at \(\frac{4}{3}\), y-intercept at \(\frac{4}{7}\). Vertical asymptote: \(x = -7\). Horizontal asymptote: \(y = -3\).

Step 1: Find the X-Intercept

To find the x-intercept, set the numerator equal to zero and solve for \(x\). \[ 4 - 3x = 0 \]Solve for \(x\): \[ x = \frac{4}{3} \]The x-intercept is \(\left(\frac{4}{3}, 0\right)\).

Step 2: Find the Y-Intercept

To find the y-intercept, substitute \(x = 0\) into the function and solve for \(s(x)\).\[ s(0) = \frac{4 - 3(0)}{0 + 7} = \frac{4}{7} \]The y-intercept is \(\left(0, \frac{4}{7}\right)\).

Step 3: Find the Vertical Asymptote

To find the vertical asymptote, set the denominator equal to zero and solve for \(x\). \[ x + 7 = 0 \]Solve for \(x\): \[ x = -7 \]There is a vertical asymptote at \(x = -7\).

Step 4: Find the Horizontal Asymptote

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. Both the numerator and the denominator are degree 1. The horizontal asymptote is the ratio of the leading coefficients:\[ y = \frac{-3}{1} = -3 \]Therefore, there is a horizontal asymptote at \(y = -3\).

Step 5: Determine the Domain

The domain of the function is all real numbers except where the denominator is zero.Since there is a vertical asymptote at \(x = -7\), the domain is \(x eq -7\).

Step 6: Determine the Range

The range is all real numbers except where the function approaches the horizontal asymptote. Since there is a horizontal asymptote at \(y = -3\), the range is \(y eq -3\).

Step 7: Sketch the Graph

Using the intercepts and asymptotes, sketch the graph:1. Plot the x-intercept at \(\left(\frac{4}{3}, 0\right)\)2. Plot the y-intercept at \(\left(0, \frac{4}{7}\right)\)3. Draw a vertical line at \(x = -7\) for the vertical asymptote4. Draw a horizontal line at \(y = -3\) for the horizontal asymptote5. The graph will cross the x-axis at \(\left(\frac{4}{3}, 0\right)\) and approach the asymptotes in each direction.

Key Concepts

X-Intercepts

To find the x-intercepts of a rational function like \(s(x) = \frac{4-3x}{x+7}\), we focus on the numerator. Intercepts occur where the function crosses the x-axis, meaning the output is zero. Thus, we need to set the numerator equal to zero: \(4 - 3x = 0\). Solving this gives us \(x = \frac{4}{3}\). This means the x-intercept is at the point \(\left(\frac{4}{3}, 0\right)\).

Remember, x-intercepts only exist where the numerator can be zero without making the denominator zero (as that would make the function undefined). Hence, x-intercepts are crucial for understanding where the graph of the function actually touches or crosses the x-axis.

Y-Intercepts

Y-intercepts are points where the function crosses the y-axis. For a rational function like \(s(x) = \frac{4-3x}{x+7}\), this happens when \(x = 0\). We substitute 0 for \(x\) in the function to find the y-intercept: \(s(0) = \frac{4 - 3(0)}{0 + 7} = \frac{4}{7}\). Therefore, the y-intercept of this function is \(\left(0, \frac{4}{7}\right)\).

The y-intercept tells us where the graph intersects the y-axis. Unlike x-intercepts, a rational function can only have one y-intercept because it is a function (single output value for every input value).

Vertical Asymptotes

A vertical asymptote occurs where the rational function is undefined, often due to division by zero in the denominator. For \(s(x) = \frac{4-3x}{x+7}\), set the denominator to zero: \(x + 7 = 0\). Solving for \(x\) gives us the vertical asymptote at \(x = -7\).

A vertical asymptote is a vertical line where the function approaches infinity or negative infinity. The graph will never touch or cross this line, but it will get infinitely close. Understanding where these asymptotes occur helps determine the behavior of the function as the input values approach these critical points.

Horizontal Asymptotes

Horizontal asymptotes reveal the end behavior of a rational function as \(x\) approaches positive or negative infinity. They are found by comparing the degrees of the numerator and denominator. For \(s(x) = \frac{4-3x}{x+7}\), both are degree 1. Thus, the horizontal asymptote is the ratio of the leading coefficients: \(\frac{-3}{1} = -3\), or at \(y = -3\).

A horizontal asymptote, unlike a vertical one, can be crossed by the graph at central points but still dictates how the graph behaves at extremes. For larger values of \(x\), the function values get closer and closer to this line but do not actually reach it. This guides us in sketching rational functions.

Domain and Range

The domain of a rational function is all real numbers except where the denominator is zero. For \(s(x) = \frac{4-3x}{x+7}\), the domain is all \(x\) except \(-7\), where the division by zero occurs, so we express this as \(x eq -7\).

The range is influenced by horizontal asymptotes. As \(s(x)\) approaches \(-3\) for very large or small \(x\) values, the range excludes \(-3\). Hence, the range is all real numbers except \(y = -3\). Understanding domain and range helps predict how the graph stretches across both axes. Recognizing these sets is important for identifying how rational functions behave throughout their entirety.