Problem 5 - Algebra Form and Function

Question

In Exercises 5-6, evaluate the function for \(x=-7\). $$ f(x)=x / 2-1 $$

Step-by-Step Solution

Answer

Answer: The value of the function \(f(-7) = \frac{-9}{2}\).

Step 1: Plug in the value of x

Substitute \(x = -7\) into the function. So, $$ f(-7) = \frac{-7}{2} - 1 $$

Step 2: Simplify expression

Now, simplify the expression: $$ f(-7) = \frac{-7}{2} - \frac{2}{2} $$ Combine the fractions by adding the numerators: $$ f(-7)= \frac{-7 - 2}{2} = \frac{-9}{2} $$

Step 3: Write the final answer

The function \(f(x) = \frac{x}{2} - 1\) evaluated at \(x = -7\) is: $$ f(-7) = \frac{-9}{2} $$

Key Concepts

Substitute Variable

To evaluate a function at a specific value, the first step is to substitute the given variable with the provided number. This process involves replacing every instance of the variable in the function with the number. In our example, the function is:\[f(x) = \frac{x}{2} - 1\]We need to determine the value of the function when \(x = -7\). By substituting \(-7\) for \(x\) in the function, we rewrite it as:\[f(-7) = \frac{-7}{2} - 1\]Substituting variables is a cornerstone of function evaluation. It helps us transform an abstract expression into a concrete number.

• Identify the variable to substitute.
• Replace it with the given value in all occurrences.

Substitution makes the problem more straightforward by converting it to basic arithmetic.

Simplify Expression

Once the variable is substituted, the next step is to simplify the expression. Simplification involves transforming the expression into its simplest form. This may include performing operations like addition, subtraction, multiplication, or division.In this problem, after substituting \(x = -7\), our expression is:\[f(-7) = \frac{-7}{2} - 1 \]To simplify this expression, we first ensure all terms have a common denominator. The number \(-1\) can be rewritten as \(-\frac{2}{2}\) to match the denominator of the fraction \(\frac{-7}{2}\):\[f(-7) = \frac{-7}{2} - \frac{2}{2}\]Simplifying requires attention to detail, ensuring all terms are correctly transformed and combined efficiently, leading to the next step.

Combine Fractions

When working with fractions, especially in an expression, it often becomes necessary to combine them. This is feasible when the fractions have a common denominator. In our situation, we have:\[f(-7) = \frac{-7}{2} - \frac{2}{2}\]Since both fractions have the same denominator, we can simply combine the numerators and retain the common denominator:\[f(-7) = \frac{-7 - 2}{2}\]Perform the arithmetic in the numerator:\[f(-7) = \frac{-9}{2}\]Combining fractions helps in reducing complex computations into simpler ones, as seen in this example. When fractions share a denominator, the process is straightforward:

• Add or subtract the numerators depending on the operation.
• Keep the common denominator consistent.

This approach punches through arithmetic obstacles, rendering the expression in its final simplified form.