Math Problem Statement

Given that f(x) = x^2 - 5 and g(x) = -2x + 3, find (f/g)(3/2), if it exists.

Solution

The problem asks to find (fg)(32)\left( \frac{f}{g} \right) \left( \frac{3}{2} \right), given the functions:

f(x)=x25f(x) = x^2 - 5 g(x)=2x+3g(x) = -2x + 3

Step 1: Define the function fg(x)\frac{f}{g}(x)

The expression fg(x)\frac{f}{g}(x) means: f(x)g(x)=x252x+3\frac{f(x)}{g(x)} = \frac{x^2 - 5}{-2x + 3}

Step 2: Substitute x=32x = \frac{3}{2} into f(x)f(x) and g(x)g(x)

  • First, calculate f(32)f\left( \frac{3}{2} \right): f(32)=(32)25=945=94204=114f\left( \frac{3}{2} \right) = \left( \frac{3}{2} \right)^2 - 5 = \frac{9}{4} - 5 = \frac{9}{4} - \frac{20}{4} = \frac{-11}{4}

  • Next, calculate g(32)g\left( \frac{3}{2} \right): g(32)=2(32)+3=3+3=0g\left( \frac{3}{2} \right) = -2\left( \frac{3}{2} \right) + 3 = -3 + 3 = 0

Step 3: Check the result

Since g(32)=0g\left( \frac{3}{2} \right) = 0, the function fg(x)\frac{f}{g}(x) is undefined at x=32x = \frac{3}{2}. Division by zero occurs, meaning fg(32)\frac{f}{g}\left( \frac{3}{2} \right) does not exist.

Conclusion:

The expression does not exist because g(32)=0g\left( \frac{3}{2} \right) = 0.

Would you like further details or have any questions?


Here are 5 related questions for practice:

  1. What happens if we try to evaluate fg(x)\frac{f}{g}(x) for other values of xx?
  2. How do you determine where a function like fg(x)\frac{f}{g}(x) is undefined?
  3. Can you find the domain of fg(x)\frac{f}{g}(x)?
  4. What is the behavior of the function fg(x)\frac{f}{g}(x) as xx approaches 32\frac{3}{2}?
  5. How can you use limits to analyze functions with undefined points?

Tip: Always check the denominator before substituting values into rational functions to avoid dividing by zero.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Functions
Division by Zero

Formulas

f(x) = x^2 - 5
g(x) = -2x + 3
Rational function: (f/g)(x) = f(x)/g(x)

Theorems

Undefined functions when dividing by zero

Suitable Grade Level

Grades 10-12