Math Problem Statement

Find the range of 3x−2

x+5

Solution

To find the range of the function f(x)=3x2x+5f(x) = \frac{3x - 2}{x + 5}, we need to determine all possible values that the function can take for different values of xx.

Step 1: Identify Domain Restrictions

First, the function is undefined when the denominator is zero, i.e., when:

x+5=0    x=5x + 5 = 0 \implies x = -5

Thus, the function is undefined at x=5x = -5, and the domain is x(,5)(5,)x \in (-\infty, -5) \cup (-5, \infty).

Step 2: Analyze the Horizontal Asymptote

To find the horizontal asymptote, consider the behavior of the function as xx \to \infty or xx \to -\infty. Divide both the numerator and denominator by xx:

f(x)=3x2x+5=32x1+5xf(x) = \frac{3x - 2}{x + 5} = \frac{3 - \frac{2}{x}}{1 + \frac{5}{x}}

As xx \to \infty or xx \to -\infty, the terms 2x\frac{2}{x} and 5x\frac{5}{x} approach zero, so:

f(x)31=3f(x) \to \frac{3}{1} = 3

Thus, the function has a horizontal asymptote at y=3y = 3.

Step 3: Check for Critical Points

To find the critical points, set f(x)=yf(x) = y and solve for xx in terms of yy:

y=3x2x+5y = \frac{3x - 2}{x + 5}

Multiply both sides by x+5x + 5:

y(x+5)=3x2y(x + 5) = 3x - 2

Expand both sides:

yx+5y=3x2yx + 5y = 3x - 2

Now, isolate xx:

yx3x=5y2yx - 3x = -5y - 2

Factor out xx:

x(y3)=5y2x(y - 3) = -5y - 2

Solve for xx:

x=5y2y3x = \frac{-5y - 2}{y - 3}

Step 4: Exclude Values for Which x=5x = -5

The function is undefined at x=5x = -5. Set the denominator equal to zero to find which yy-value is not possible:

y3=0    y=3y - 3 = 0 \implies y = 3

Thus, y=3y = 3 is not in the range, since f(x)3f(x) \to 3 as xx \to \infty, but it never equals 3.

Conclusion: The Range

The range of the function is all real numbers except y=3y = 3:

(,3)(3,)\boxed{(-\infty, 3) \cup (3, \infty)}

Would you like more details on this process?


Here are 5 related questions to explore further:

  1. What is the domain of the function f(x)=3x2x+5f(x) = \frac{3x - 2}{x + 5}?
  2. How do you find the vertical asymptote of a rational function?
  3. How do you graph a rational function and identify its asymptotes?
  4. What is the general method to find the range of any rational function?
  5. How does the range change if the numerator and denominator have higher degrees?

Tip: Always check for asymptotes (vertical and horizontal) when analyzing rational functions, as they provide insight into the function's range and behavior.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Rational Functions
Asymptotes

Formulas

f(x) = (3x - 2) / (x + 5)
Horizontal Asymptote: y = 3
x = (-5y - 2) / (y - 3)

Theorems

Asymptote Theorem: Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity.

Suitable Grade Level

Grades 10-12