Problem 45 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=\frac{x-1}{x}$$

Step-by-Step Solution

Answer

Graph features a vertical asymptote at \( x=0 \), x-intercept at \( x=1 \), approaching \( y=1 \) as \( x \to \pm \infty \).

Step 1: Rewrite the Function

Begin by rewriting the function for clarity. The function given is \( f(x) = \frac{x-1}{x} \). This can be expressed as two separate terms: \( f(x) = 1 - \frac{1}{x} \). This separation will make it easier to graph.

Step 2: Identify Key Features

Identifying key features such as intercepts and asymptotes is crucial. For the y-intercept, set \( x = 0 \) in the equation; however, notice that \( x=0 \) causes the expression \( \frac{1}{x} \) to be undefined, indicating a vertical asymptote at \( x = 0 \). For the x-intercept, set \( f(x) = 0 \), resulting in \( 1 - \frac{1}{x} = 0 \). Solving this gives \( x = 1 \) as the x-intercept. There are no horizontal asymptotes for this rational function as long as \( x eq 0 \).

Step 3: Sketch the Asymptote and Intercepts

Draw the vertical asymptote at \( x = 0 \). Mark the x-intercept at \( (1, 0) \). Since the y-intercept does not exist due to the vertical asymptote, note that the graph gets closer to \( x = 0 \) without intersecting it.

Step 4: Analyze and Plot the Graph

Consider how the function behaves as \( x \) approaches the asymptotes. As \( x \to 0^+ \), \( f(x) \to -\infty \); as \( x \to 0^- \), \( f(x) \to +\infty \). For \( x > 0 \), \( f(x) \to 1 \) as \( x \to \infty \) and for \( x < 0 \), \( f(x) \to 1 \) as \( x \to -\infty \). Plot points around the x-intercept to see the curve of the graph around \( x = 1 \).

Step 5: Draw the Graph

Using the intercept and asymptotic behavior, sketch the graph. It should swoop from the top left to bottom right through the x-intercept, staying close to the horizontal line \( y = 1 \) as \( x \to \pm \infty \), and sharply turn about the vertical asymptote at \( x = 0 \).

Key Concepts

Vertical Asymptotes

In the world of graphing rational functions, vertical asymptotes are lines that the graph of a function approaches but never touches. They occur where the denominator of a rational function is zero and the numerator isn't zero as well. For the function \( f(x) = \frac{x - 1}{x} \), analyze where it is undefined. The denominator is zero at \( x = 0 \).This means we have a vertical asymptote at \( x = 0 \).This tells us that no matter the value of \( x \), it never actually reaches zero, but gets infinitely close. When you graph this, you'll notice that on either side of \( x = 0 \), the function goes towards infinity or negative infinity.

• As \( x \to 0^+ \) (approaching from the right), \( f(x) \to -\infty \).
• As \( x \to 0^- \) (approaching from the left), \( f(x) \to +\infty \).

These behaviors are essential clues in sketching the graph of the function. They indicate where the curves sharply change direction and why an asymptote is present in that location.

X-Intercepts

Finding x-intercepts is crucial when graphing a rational function.This is where the graph crosses the x-axis, meaning the output or \( f(x) \) is zero. To find the x-intercept of our function, set the numerator equal to zero and solve.For \( f(x) = \frac{x - 1}{x} \), set the numerator equal to zero: \( x - 1 = 0 \).Solving this gives \( x = 1 \).So, our x-intercept is located at the point \( (1, 0) \).The significance of the x-intercept is visual and analytical:

• It marks a point where the function takes a brief pause at zero before diving or rising towards the asymptote or other parts of the graph.
• Connecting the x-intercept with the graph helps shape the curve and provides important transitional behavior.

Understanding x-intercepts helps effectively plot the function and anticipate the structure of the graph.

Graph Behavior

Examining the behavior of the graph of a rational function provides insights into its overall shape.In our function \( f(x) = \frac{x - 1}{x} \), analyzing both the x-intercepts and vertical asymptotes helps formulate graph behavior.As \( x \) approaches the vertical asymptote \( x = 0 \), our function rises to \( +\infty \) from the left and descends to \(-\infty\) from the right.This sharp turn-around is indicative of behavior influenced by vertical asymptotes.For \( x \to \pm\infty \), notice how \( f(x) \to 1 \). The function approaches the horizontal line \( y = 1 \), which can act like a boundary.This behavior signifies that far away from the axis, the graph levels off and doesn't rise or fall sharply. Plotting the graph involves:

• Marking the vertical asymptote, keeping in mind how it affects nearby graph sections.
• Identifying the x-intercept, noting the brief pause as the graph crosses the x-axis.
• Sketching the curve by considering how the function approaches the asymptote and distant behavior toward \( y = 1 \).

These points help predict how the graph moves, portraying a comprehensive picture of the rational function's behavior.