Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 9

Chapter 4, Problem 9

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.

As $x \to - 3^{-}$ , $f (x) \to$ ____

Verified step by step guidance

Identify the point of interest on the x-axis, which is $x \to -3^-$, meaning we are approaching $-3$ from the left side.

Look at the graph near $x = -3$ on the left side to observe the behavior of the function $f(x)$ as $x$ approaches $-3$ from values less than $-3$.

Notice the value of $f(x)$ as $x$ gets closer to $-3$ from the left. Check if the function values increase without bound (go to $+\infty$), decrease without bound (go to $-\infty$), or approach a finite number.

From the graph, observe that as $x$ approaches $-3$ from the left, the function $f(x)$ decreases without bound, meaning $f(x) \to -\infty$.

Summarize the behavior: As $x \to -3^-$, $f(x) \to -\infty$.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vertical Asymptotes

Vertical asymptotes occur where the function approaches infinity or negative infinity as the input approaches a specific value. They indicate values of x where the function is undefined, often due to division by zero in rational functions. In the graph, vertical asymptotes are shown at x = 6 and x = 14.

Horizontal Asymptotes

A horizontal asymptote represents the value that the function approaches as x tends to positive or negative infinity. It shows the end behavior of the function. In this graph, the horizontal asymptote is y = 0, meaning the function values get closer to zero as x becomes very large or very small.

Limit Behavior Near a Point

The limit of a function as x approaches a specific value from the left or right describes the function's behavior near that point. For example, as x approaches -3 from the left (x → -3⁻), the function value approaches a certain number or infinity. Understanding this helps in interpreting the graph and completing limit statements.

Recommended video:

05:01

Identifying Intervals of Unknown Behavior

Related Practice

Textbook Question

Among all pairs of numbers whose difference is 14, find a pair whose product is as small as possible. What is the minimum product?

views

Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x²+5x+4>0

views

Textbook Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2(x−3)²+1

views

Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x²−6x+9<0

views

Textbook Question

Divide using long division. State the quotient, and the remainder, $r (x)$ . $\frac{2x^3 + 7x^2 + 9x - 20}{x + 3}$

views

Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=x^1/3 −4x²+7

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.As x→−3−x\(\to\)-3^{-}, f(x)→f\(\left\)(x\(\right\))\(\to\)____

Verified video answer for a similar problem:

Key Concepts

Vertical Asymptotes

Horizontal Asymptotes

Limit Behavior Near a Point

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.

As $x \to - 3^{-}$ , $f (x) \to$ ____