Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{1}{x+3}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\frac{1}{x+3}\) has a y-intercept at (0, 1/3), a vertical asymptote at x = -3, and a horizontal asymptote at y = 0. It is not symmetric about the y-axis or the origin.

Step 1: Intercepts

Find where the function intersects the x-axis and the y-axis. The x-intercept is found when f(x) = 0. For the given function, f(x)=(1/(x+3)), the function will never equals 0, hence there are no x-intercepts. The y-intercept is found by evaluating the function at x = 0. In the given function, at x=0 the function f(x) becomes f(0)= 1/3, hence the y-intercept is at (0, 1/3).

Step 2: Symmetry

This function f(x)=(1/(x+3)) is not symmetrical around the y-axis or the origin. The function does not pass the vertical line test for symmetry about the y-axis or the origin. The test is to replace x with -x in the equation and see if the same equation results. Here this is not the case, thus the graph isn't symmetric.

Step 3: Vertical Asymptotes

A vertical asymptote is found when the function approaches infinity. For the function f(x)=(1/(x+3)), this happens when the denominator equals zero, i.e., when x+3 = 0, hence at x = -3. So, the function has a vertical asymptote at x = -3.

Step 4: Horizontal asymptotes

A function has a horizontal asymptote if, as the function approaches positive or negative infinity, the function approaches a certain constant. In the case of the function f(x)=(1/(x+3)), as x approaches positive or negative infinity, the function approaches 0. Therefore, the function has a horizontal asymptote of y = 0.

Key Concepts

Intercepts

When working with rational functions like \( f(x) = \frac{1}{x+3} \), finding intercepts helps you understand where the graph crosses the axes.

• X-intercept: This is where the graph crosses the x-axis, meaning \( f(x) = 0 \). For rational functions, the numerator must be zero for there to be an x-intercept. Since the numerator is 1 and never zero, there are no x-intercepts for this function.
• Y-intercept: This is where the graph crosses the y-axis, found by evaluating \( f(x) \) at \( x = 0 \). For \( f(x) = \frac{1}{x+3} \), plugging in \( x = 0 \) gives \( f(0) = \frac{1}{3} \). Thus, the y-intercept is at \( (0, \frac{1}{3}) \).

Understanding intercepts gives clues on how the graph is positioned relative to the axes.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually touches or crosses. They help in understanding how the graph behaves at extreme values of x.

• Vertical Asymptote: Occurs where the function is undefined. For \( f(x) = \frac{1}{x+3} \), the denominator is zero when \( x = -3 \), which creates a vertical asymptote at this point.
• Horizontal Asymptote: Describes the behavior of the graph as \( x \) approaches positive or negative infinity. For this function, as \( x \) heads toward infinity, \( f(x) \) approaches 0. Hence, it has a horizontal asymptote at \( y = 0 \).

These asymptotes guide the sketching of the graph around these lines as it indicates the behavior of the function near the asymptotes.

Graph Sketching

Graph sketching for rational functions involves integrating information about intercepts and asymptotes.

• Start by plotting intercepts, as we have a y-intercept at \( (0, \frac{1}{3}) \).
• Draw the vertical asymptote at \( x = -3 \). The graph will never touch this line, instead, it will stretch towards infinity as it approaches \( x = -3 \) from both sides.
• Include the horizontal asymptote at \( y = 0 \). This indicates that as \( x \) moves far left or right, \( f(x) \) gets closer and closer to zero.

Combine these elements to best approximate the path of the graph. Generally, the graph will consist of two separate sections that hug the vertical asymptote, one above and one below the horizontal asymptote.

Symmetry

Symmetry in a graph can make sketching easier and often reveals interesting properties about the function. For \( f(x) = \frac{1}{x+3} \), checking for symmetry can help in understanding the function's balance or any pattern it may follow.

• To check for y-axis symmetry, substitute \( -x \) for \( x \) and see if the result is the original function. This doesn't hold here as \( f(-x) = \frac{1}{-x+3} \) is not equivalent to \( f(x) \).
• For origin symmetry, replacing \( x \) with \( -x \) and \( f(x) \) with \( -f(x) \) should give an equivalent function. Again, this condition is not satisfied.

Since the function does not meet these criteria, it lacks symmetry about the y-axis and the origin. This absence of symmetry means its graph won't mirror itself across these axes.