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In mathematics, **function evaluation** refers to determining the output of a function for a specific input value. When you have a function, such as \( f(x) = x^2 - 9 \), you can replace the variable \( x \) with any number to find the function's value at that number.

For instance, evaluating \( f \left( \frac{1}{2} \right) \) involves replacing \( x \) with \( \frac{1}{2} \) in the expression \( x^2 - 9 \) and calculating the result:
\[ f \left( \frac{1}{2} \right) = \left( \frac{1}{2} \right)^2 - 9 = \frac{1}{4} - 9 = -\frac{35}{4} \]
This process is essential for understanding how functions behave at specific points and is often used in calculus and algebra tasks to solve real-world problems.

The **quotient of functions** involves dividing one function by another. Given two functions \( f(x) = x^2 - 9 \) and \( g(x) = 2x \), the quotient function \( \left( \frac{f}{g} \right)(x) \) is defined by dividing the expressions for \( f(x) \) and \( g(x) \) at a specific value of \( x \).

To find the value of \( \left( \frac{f}{g} \right) \left( \frac{1}{2} \right) \), you first compute each function at \( \frac{1}{2} \) and then divide the results:
\[ \left( \frac{f}{g} \right) \left( \frac{1}{2} \right) = \frac{f \left( \frac{1}{2} \right)}{g \left( \frac{1}{2} \right)} = \frac{-\frac{35}{4}}{1} = -\frac{35}{4} \]
This operation is helpful in simplifying complex functions and solving equations where functions are combined.

In **substitution**, you replace the variable in an expression with a given number, other variables, or even another function. It's a versatile technique in algebra and calculus used for simplifying and solving equations.

Here, to find \( \left( \frac{f}{g} \right) \left( \frac{1}{2} \right) \), substitution plays a critical role. You first evaluate \( f(x) \) and \( g(x) \) by substituting \( \frac{1}{2} \) for \( x \):

• Evaluate \( f \left( \frac{1}{2} \right) \) by substituting \( \frac{1}{2} \) into \( f(x) = x^2 - 9 \):
  \[ f \left( \frac{1}{2} \right) = \left( \frac{1}{2} \right)^2 - 9 = -\frac{35}{4} \]

• Evaluate \( g \left( \frac{1}{2} \right) \) by substituting \( \frac{1}{2} \) into \( g(x) = 2x \):
  \[ g \left( \frac{1}{2} \right) = 2 \times \frac{1}{2} = 1 \]

By substituting correctly, you ensure accurate results when combining functions, whether adding, subtracting, multiplying, or dividing them.