Problem 10 - Precalculus: Functions and Graphs

Question

Sketch the graph of \(f\) $$f(x)=\frac{4 x}{2 x-5}$$

Step-by-Step Solution

Answer

The graph has a vertical asymptote at \(x = \frac{5}{2}\) and a horizontal asymptote at \(y = 2\), passing through the origin.

Step 1: Determine the Domain

The function is defined for all values of \(x\) except where the denominator is zero. Set the denominator equal to zero and solve for \(x\): \[ 2x - 5 = 0 \] \[ x = \frac{5}{2} \]. Thus, the domain of \(f\) is all real numbers except \(x = \frac{5}{2}\).

Step 2: Find the Vertical Asymptote

From Step 1, we know that the function is not defined at \(x = \frac{5}{2}\). This indicates a vertical asymptote at \(x = \frac{5}{2}\).

Step 3: Find the Horizontal Asymptote

The horizontal asymptote is found by considering the behavior as \(x\) approaches infinity. The degrees of the numerator and denominator are both 1, so the horizontal asymptote is found by dividing the leading coefficients: \[ y = \frac{4}{2} = 2 \]. Thus, there is a horizontal asymptote at \(y = 2\).

Step 4: Find the Intercepts

To find the y-intercept, set \(x = 0\): \[ f(0) = \frac{4 \times 0}{2 \times 0 - 5} = 0 \]. Thus, the y-intercept is at (0,0).To find the x-intercept, set \(f(x) = 0\): \[ \frac{4x}{2x-5} = 0 \Rightarrow x = 0 \]. Thus, the x-intercept is also at (0,0).

Step 5: Plot Key Points and Asymptotes

Plot the intercept (0,0) and draw the vertical asymptote at \(x = \frac{5}{2}\) and the horizontal asymptote at \(y = 2\).

Step 6: Sketch the Graph

Using the information gathered, sketch the graph. The function will approach the horizontal asymptote \(y = 2\) as \(x\) goes towards positive or negative infinity, and it will approach the vertical asymptote \(x = \frac{5}{2}\) without touching it. The curve should pass through the intercept (0,0).

Key Concepts

Domain of a Function

The domain of a function is all the values that the independent variable (usually x) can take so that the function itself is defined. For rational functions like \( f(x) = \frac{4x}{2x-5} \), the domain is determined by identifying values that make the denominator zero, since division by zero is undefined. This means we need to set the denominator equal to zero and solve for x: \( 2x - 5 = 0 \). Therefore, \( x = \frac{5}{2} \) is the value where the function is not defined.
Thus, the domain of \( f(x) \) is all real numbers except \( x = \frac{5}{2} \). To express this clearly, we write it as:

• \( x \in \mathbb{R}, x eq \frac{5}{2} \)

Identifying the domain is crucial as it reveals restrictions on the input values and helps in understanding the behavior of the function.

Vertical Asymptotes

Vertical asymptotes occur at values of x where the function tends toward infinity, and these are typically the values that make the denominator zero in a rational function. In our example, \( f(x) = \frac{4x}{2x-5} \), we identified \( x = \frac{5}{2} \) when analyzing the domain. This point indicates a vertical asymptote.
What does this mean?

• As x approaches \( \frac{5}{2} \) from either side, the value of \( f(x) \) increases or decreases without bound.
• The graph will never actually cross the line \( x = \frac{5}{2} \).

Understanding vertical asymptotes is essential as they indicate where a function behaves infinitely and help visualize the function's behavior around certain critical points.

Horizontal Asymptotes

Horizontal asymptotes portray the behavior of a function as x approaches infinity or negative infinity. For this function, \( f(x) = \frac{4x}{2x-5} \), the degrees of the numerator and denominator are equal (each has degree 1). The horizontal asymptote can be found by dividing the leading coefficients: \( \frac{4}{2} = 2 \). Thus, the horizontal asymptote is \( y = 2 \).
What does this tell us?

• As x becomes very large or very small, the graph of f(x) approaches the line \( y = 2 \).
• This asymptote represents a leveling-out point for the function in the extremes.

Horizontal asymptotes help us understand the end behavior of functions and are essential tools for sketching the long-term trend of the graph.

Intercepts of a Function

The intercepts of a function are where the graph crosses the x-axis and y-axis. For rational functions like \( f(x) = \frac{4x}{2x-5} \), finding these points provides key information about its graph.
**Y-intercept:**

• Set \( x = 0 \) and solve for \( f(x) \).
• \( f(0) = \frac{4 \times 0}{2 \times 0 - 5} = 0 \).
• Thus, the y-intercept is at (0,0).

**X-intercept:**

• Set \( f(x) = 0 \) to find where the numerator equals zero.
• \( \frac{4x}{2x-5} = 0 \Rightarrow x = 0 \).
• Thus, the x-intercept is also at (0,0).

Locating and understanding intercepts is crucial as they serve as starting points for graphing and provide insight into the interplay between the function's variables.