To determine the range of the function $f(x) = \frac{x-5}{x+2}$, we analyze the function thoroughly. Here's a step-by-step breakdown:

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1. Understanding the Function

The function $f(x) = \frac{x-5}{x+2}$ is a rational function, where both the numerator and denominator are linear polynomials.

• The denominator $x+2$ cannot be zero. Thus, $x \neq -2$.

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2. Horizontal Asymptote (Behavior as $x \to \infty$ or $x \to -\infty$)

When the degree of the numerator and denominator is the same (as in this case), the horizontal asymptote is given by the ratio of the leading coefficients.

• In $f(x) = \frac{x-5}{x+2}$, the leading coefficient of $x$ in both the numerator and denominator is 1.
  Thus, as $x \to \infty$ or $x \to -\infty$, $f(x) \to 1$.

So, there is a horizontal asymptote at $y = 1$.

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3. Range of the Function

The horizontal asymptote at $y = 1$ tells us that the function approaches but never equals $1$. This suggests that $y = 1$ is excluded from the range.

To confirm this, we solve for $x$ in terms of $y$ to find the set of all possible $y$-values:

$y = \frac{x-5}{x+2}.$

Multiply through by $x+2$ (assuming $x \neq -2$):

$y(x+2) = x-5.$

Expand and isolate $x$:

$yx + 2y = x - 5.$

Rearranging terms to isolate $x$-terms on one side:

$yx - x = -2y - 5.$

Factor out $x$ on the left:

$x(y-1) = -2y - 5.$

Solve for $x$:

$x = \frac{-2y - 5}{y - 1}, \quad \text{where } y \neq 1.$

The expression for $x$ is defined for all $y \neq 1$. Therefore, the range of $f(x)$ is:

$\boxed{(-\infty, 1) \cup (1, \infty)}.$

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Final Answer:

The range of $f(x) = \frac{x-5}{x+2}$ is all real numbers except $y = 1$, or:

$\boxed{(-\infty, 1) \cup (1, \infty)}.$

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Clarification:

• The function never takes the value $y = 1$ due to the horizontal asymptote.
• The rest of the real numbers are part of the range.

Would you like further clarification or a graph of the function for better understanding? Let me know!

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Related Questions:

1. What is the domain of $f(x) = \frac{x-5}{x+2}$?
2. How do you find vertical asymptotes of $f(x) = \frac{x-5}{x+2}$?
3. What happens when $x \to -2$ in $f(x) = \frac{x-5}{x+2}$?
4. How can we graph $f(x) = \frac{x-5}{x+2}$ and determine its behavior?
5. What is the range of $f(x) = \frac{ax+b}{cx+d}$ in general?

Tip:

To determine the range of a rational function, solve for $x$ in terms of $y$ and check for values of $y$ that make $x$ undefined.