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

\[
f(x) = x^2 - 5
\]
\[
g(x) = -2x + 3
\]

### Step 1: Define the function \(\frac{f}{g}(x)\)
The expression \(\frac{f}{g}(x)\) means:
\[
\frac{f(x)}{g(x)} = \frac{x^2 - 5}{-2x + 3}
\]

### Step 2: Substitute \(x = \frac{3}{2}\) into \(f(x)\) and \(g(x)\)
- First, calculate \(f\left( \frac{3}{2} \right)\):
\[
f\left( \frac{3}{2} \right) = \left( \frac{3}{2} \right)^2 - 5 = \frac{9}{4} - 5 = \frac{9}{4} - \frac{20}{4} = \frac{-11}{4}
\]

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

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

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

Would you like further details or have any questions?

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Here are 5 related questions for practice:
1. What happens if we try to evaluate \(\frac{f}{g}(x)\) for other values of \(x\)?
2. How do you determine where a function like \(\frac{f}{g}(x)\) is undefined?
3. Can you find the domain of \(\frac{f}{g}(x)\)?
4. What is the behavior of the function \(\frac{f}{g}(x)\) as \(x\) approaches \(\frac{3}{2}\)?
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.

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

Learn how to evaluate the rational function (f/g)(3/2), given f(x) = x^2 - 5 and g(x) = -2x + 3. Understand why the function is undefined at x = 3/2 due to division by zero. Suitable for Grades 10-12.

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