A rectangular box has length, width, and height that are consecutive whole numbers. The volume is 120 cubic inches. Which set gives the correct dimensions in inches?

A rectangular box has length one inch more than the width, which is one inch more than the height. The volume is 86.625 cubic inches. What is the height in inches?

A rectangular box has length three times the height, and the height is one inch less than the width. The volume is 108 cubic inches. Find the width in inches.

A right circular cylinder has radius 3 inches more than the height. The volume is 16π cubic inches. What is the height? Use $$V = \pi r^2 h$$ .

A right circular cylinder’s radius and height differ by one meter; the radius is larger. The volume is 48π cubic meters. Which ordered pair (r, h) in meters satisfies V = $$\pi r^2 h$$ ?

A right circular cylinder’s radius and height differ by two meters, with the height greater than the radius. The volume is 28.125π cubic meters. Find the height in meters. Use $$V = \pi r^2 h$$ .

Average cost function for producing x items is f(x) = $$(15,000x − 0.1x^2 + 1000)/x$$ . Which expression simplifies f(x) correctly for x > 0?

Which statement best defines a rational function in this section’s context?

The average cost function for producing x items is given by f(x) = $$(15,000x - 0.1x^2 + 1000) / x$$ . What is a direct implication of the variable x being in the denominator?

Consider the reciprocal function f(x) = 1/x. Based on the description provided, which statement about its local behavior near x = 0 is correct?

Arrow notation summarizes limits and asymptotic behavior. Match the symbol to its meaning: $$x \to a^+$$ , $$x \to a^−$$ , $$f(x) \to \infty$$ , $$f(x) \to a$$ .

Using arrow notation and the given observations, which statement about f(x) = 1/x is correct?

Consider the reciprocal function f(x) = 1/x. Using the local behavior near x = 0 shown, which statement correctly describes the function values as x approaches 0 from the left?

For f(x) = 1/x, what happens to f(x) as x approaches 0 from the right? Use the observed values 0.1, 0.01, 0.001, 0.0001.

Which statement best describes the end behavior of f(x) = 1/x as x → ±∞, based on the plotted graph?

A vertical asymptote of a graph is defined as a vertical line x = a where the graph tends toward positive or negative infinity as x approaches a from either side. Which notation captures this behavior?

Using the arrow notation for f(x) = 1/x, select the correct pair describing behavior near x = 0 and for large |x|.

Define a horizontal asymptote using the formal description provided and choose the expression that correctly uses arrow notation for a function with horizontal asymptote y = b.

The graph in Figure 6 shows a vertical dashed line at x = 2 and a horizontal dashed line at y = 4. Use arrow notation to state the local behavior near the vertical asymptote.

For the function graphed in Figure 6, identify the horizontal asymptote and write the end behavior using arrow notation.

For the function f(x) = 1/(x+2) + 3, choose the correct end behavior statement.

The transformed rational function in Example 2 can be written equivalently as f(x) = (3x + 7)/(x + 2). Which statement about its asymptotes matches the analysis?

A rational function is defined as a function that can be written as the quotient of two polynomials. Select the option that best matches this definition.

Recall the definition: A rational function is any function that can be written as the quotient of two polynomials P(x) and Q(x) with Q(x) ≠ 0. Which expression represents a rational function?

Given the concentration model C(t) = (5 + t)/(100 + 10t), identify the horizontal asymptote of C(t) and interpret it in context.

Recall the rule for determining the domain of a rational function. Which statement correctly describes the domain?

Consider the graph of the rational function f(x) = $$(x + 3) / (x^2 − 9)$$ . Based on the domain analysis, which feature appears at x = 3?

A removable discontinuity of a rational function occurs at x = a if a is a zero for which kind of factor relationship between numerator and denominator?

Graph-based interpretation: A plotted rational function shows a dashed vertical line at x = −2 and another at x = 1, with the curve approaching these lines. What conclusion can you draw about the function’s behavior at x = −2 and x = 1?

Recall the rule: When the degree of the denominator is greater than the degree of the numerator for a rational function f(x)=p(x)/q(x), what is the horizontal asymptote?

A rational function has numerator degree one larger than the denominator degree. Which end-behavior feature occurs?

Given f(x) = $$\frac{4x + 2}{x^2 + 4x - 5}$$ , determine its horizontal asymptote based on degree comparison and end behavior.

For f(x) = $$(3x^2 − 2x + 1)/(x − 1)$$ , identify the slant asymptote by performing polynomial division and interpreting end behavior.

For f(x) = $$\frac{3x^2 + 2}{x^2 + 4x - 5}$$ , determine the horizontal asymptote using the ratio of leading coefficients.

A rational function’s graph may cross a horizontal asymptote but will never cross which type of asymptote?

Consider g(x) = $$(6x^3 - 10x) / (2x^3 + 5x^2)$$ . Identify its horizontal asymptote using leading coefficients.

For h(x) = $$(x^2 − 4x + 1)/(x + 2)$$ , find the slant asymptote by division. Use the given quotient and remainder information.

Determine the horizontal asymptote of $$k(x) = \frac{x^2 + 4x}{x^3 - 8}$$ by comparing degrees.

Recall the rule for horizontal asymptotes of a rational function when the degrees of the numerator and denominator are equal. Which statement correctly identifies the horizontal asymptote for C(t) = (5 + t) / (100 + 10t)?

Analyze the end behavior of f(x) = ((x − 2)(x + 3)) / ((x − 1)(x + 2)(x − 5)). Which statement best describes its horizontal asymptote and why?

A rational function will have a y-intercept when which condition is met? Choose the best statement.

Which inputs can produce x-intercepts for a rational function?

In the graph of a rational function, vertical asymptotes correspond to which features of the function?

When a factor in the denominator has odd degree, how does the graph behave near the corresponding vertical asymptote?

Consider the parent rational function y = 1/x. Describe the end behavior near the vertical asymptote x = 0 and the horizontal asymptote y = 0. Choose the statement that best matches the graph: As x approaches 0 from the right and from the left, and as x → ±∞.

A rational function has a denominator factor with even multiplicity producing a vertical asymptote. Which statement best describes the distinguishing characteristic of the graph's behavior near that asymptote?

For f(x) = $$((x + 1)^2 (x − 3)) / ((x + 3)^2 (x − 2))$$ , which point is an x-intercept where the graph bounces (touches and turns) rather than crosses?

According to the 'Given a rational function, sketch a graph' procedure, which step should be performed first?

For g(x) = $$\frac{(x + 2)(x - 3)}{(x + 1)^2 (x - 2)}$$ , which statement about intercepts is correct?

A rational function has x-intercepts at x = x1, x2, …, xn, vertical asymptotes at x = v1, v2, …, vm, and no horizontal asymptote other than y = 0. Which general form correctly expresses f(x) using factors from intercepts and asymptotes?

From the graph of a rational function, the curve crosses the x-axis at x = 3 and just touches and turns at x = −2. What multiplicities do these behaviors suggest for the corresponding factors in the numerator?

Suppose a graph shows vertical asymptotes at x = −1 and x = 2. Near x = −1, the branches go to opposite infinities on either side. Near x = 2, both sides head to negative infinity. Which choice best matches the simplest denominator consistent with this behavior?

A graph of a rational function has an x-intercept at x = 1 where the curve crosses the axis, and vertical asymptotes at x = −1 and x = 2. Which statement is correct about the factors and their multiplicities?

Using the “HOW TO” guidance for writing a rational function from a graph, which step is used to determine the stretch factor a?

Consider f(x) = a (x + 2)(x − 3)/[(x + 1)(x − 2)2]. If the graph passes through the y-intercept (0, −2), what is the value of a?

What is a fundamental difference between typical graphs of polynomial functions and rational functions regarding continuity and asymptotic behavior?

If the graph of a rational function has a removable discontinuity at x=a, what must be true of the functional rule at x=a?

Find the domain of f(x)= $$\frac{x−1}{x+2}$$ . State excluded x-values.

Find the domain of f(x)= $$\frac{x+1}{x^2−1}$$ .

Find the domain of f(x)= $$\frac{x^2+4}{x^2−2x−8}$$ .

A rational function has vertical asymptotes at x = −4 and x = −1, x-intercepts at (1, 0) and (5, 0), and y-intercept at (0, 7). Which of the following is a correct equation?

From the graph, identify the vertical asymptotes. The graph displays dashed lines at x = −2 and x = 4 with curves approaching these lines.

A graph shows vertical asymptotes at x = −3 and x = 5. The y-intercept is y = 0. Which equation below is consistent with these features and also has an x-intercept at x = 2?

Graph shown has a vertical asymptote marked with an orange dashed line at x = 2. The curve approaches +∞ on the right of the asymptote and −∞ on the left, and passes through the origin, increasing and flattening for large x. Based on this visual, which statement best describes the end behavior as x → ∞?

In the graph with an orange dashed vertical line at x = −2, the function is decreasing across the right half-plane and approaches 0 as x increases. As x approaches −2 from the right, the function tends to +∞. What is the sign of f(x) immediately to the left of the vertical asymptote x = −2?

Consider the graph where the orange dashed vertical asymptote is at x = −3. The left branch is in Quadrant II approaching +∞ near the asymptote, and the right branch is in Quadrant IV approaching −∞ near the asymptote and tending to 0 for large positive x. Which interval contains x-values where f(x) > 0?

The displayed graph shows two vertical asymptotes (orange dashed) at x = 3 and x = 5. The curve has peaks near each asymptote and approaches 0 between and outside them. Which statement is most consistent with this behavior?

For f(x) = 1/(x − 2), make a small numeric table to analyze behavior near the vertical asymptote and horizontal asymptote. Which row correctly shows values near x = 2?

For f(x) = 2x/(x + 4), which statement about the sign of f(x) is true?

Consider f(x) = 2/(x + 1). Using a graphing calculator, solve f(x) > 0. Which solution set is correct?

Let f(x) = 4/(2x − 3). Using the graph, for which x is f(x) > 0?

For the function f(x) = $$(x^3 + 1)/(x + 1)$$ , what is the x-value of the removable discontinuity (hole)?

For the function f(x) = $$(x^2 + x − 6)/(x − 2)$$ , which value of x creates a removable discontinuity?