$f(x)=\frac{(x-2)(x+1)}{x+1}$

Which statements describe the end behavior of the graph of the function shown? Check all that apply.

As $x \rightarrow \infty, y \rightarrow 1$.
As $x \rightarrow \infty, y \rightarrow \infty$.
As $x \rightarrow \infty, y \rightarrow-\infty$.
As $x \rightarrow-\infty, y \rightarrow 1$.
As $x \rightarrow-\infty, y \rightarrow \infty$.
As $x \rightarrow-\infty, y \rightarrow-\infty$.

Question

05-11-2026
Mathematics

Answered

$f(x)=\frac{(x-2)(x+1)}{x+1}$

Which statements describe the end behavior of the graph of the function shown? Check all that apply.

As $x \rightarrow \infty, y \rightarrow 1$.
As $x \rightarrow \infty, y \rightarrow \infty$.
As $x \rightarrow \infty, y \rightarrow-\infty$.
As $x \rightarrow-\infty, y \rightarrow 1$.
As $x \rightarrow-\infty, y \rightarrow \infty$.
As $x \rightarrow-\infty, y \rightarrow-\infty$.

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T17:30:21-05:00

Simplify the function $f(x) = \frac{(x - 2)(x + 1)}{x + 1}$ by canceling the factor $(x + 1)$, resulting in $f(x) = x - 2$ for $x eq -1$.
Evaluate the limit as $x \to \infty$: $\lim_{x \to \infty} (x - 2) = \infty$.
Evaluate the limit as $x \to -\infty$: $\lim_{x \to -\infty} (x - 2) = -\infty$.
The end behaviors are: $\boxed{\text{As } x \rightarrow \infty, y \rightarrow \infty; \text{ As } x \rightarrow -\infty, y \rightarrow -\infty}$.

Explanation

Simplifying the Function and Domain Analysis

The function provided is $f(x) = \frac{(x - 2)(x + 1)}{x + 1}$. To analyze its end behavior, we first identify its domain. The expression is undefined when the denominator is zero, which occurs at $x = -1$. For all other real numbers $x eq -1$, we can simplify the expression by canceling the common factor of $(x + 1)$ in the numerator and the denominator, resulting in the simplified function $f(x) = x - 2$. The graph of this function is a straight line $y = x - 2$ with a hole at the point $(-1, -3)$.

End Behavior as x Approaches Positive Infinity

To determine the end behavior as $x \to \infty$, we evaluate the limit of the simplified function as $x$ grows indefinitely large. Substituting the simplified form, we have:
\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} (x - 2) \]
As $x$ increases without bound, the term $x - 2$ also increases without bound, meaning $y$ approaches $\infty$. Therefore, the statement "As $x \to \infty, y \to \infty$" is correct.

End Behavior as x Approaches Negative Infinity

To determine the end behavior as $x \to -\infty$, we evaluate the limit of the simplified function as $x$ decreases indefinitely (i.e., becomes more and more negative). We have:
\[ \lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (x - 2) \]
As $x$ decreases without bound, the term $x - 2$ also decreases without bound, meaning $y$ approaches $-\infty$. Therefore, the statement "As $x \to -\infty, y \to -\infty$" is correct.

Conclusion of End Behavior Statements

Based on the limits calculated:

As $x \to \infty$, $y \to \infty$.
As $x \to -\infty$, $y \to -\infty$.

The correct statements are:

As $x \rightarrow \infty, y \rightarrow \infty$.
As $x \rightarrow -\infty, y \rightarrow -\infty$.

$\boxed{\text{As } x \rightarrow \infty, y \rightarrow \infty; \text{ As } x \rightarrow -\infty, y \rightarrow -\infty}$

Examples

In economics, this type of linear behavior is often used to model supply and demand relationships where price and quantity increase proportionally. For example, if a company's production cost per unit is modeled linearly, knowing the end behavior helps predict total costs as output increases, allowing for better scaling and budget planning. Understanding these limits is fundamental in ensuring business models remain sustainable as they grow.