Problem 55 - Algebra A Combined Function

Question

For each function, find the indicated values. \(f(x)=\frac{1}{2} x ;\) a. \(f(0)\) b. \(f(2)\) c. \(f(-2)\)

Step-by-Step Solution

Answer

a. 0, b. 1, c. -1

Step 1: Identify the Function

We are given the function to work with, which is \( f(x) = \frac{1}{2}x \). This means for any input \( x \), the output is obtained by multiplying \( x \) by \( \frac{1}{2} \).

Step 2: Substitute for f(0)

To find \( f(0) \), substitute \( x = 0 \) into the function: \( f(0) = \frac{1}{2} \times 0 \). Simplify to get \( f(0) = 0 \).

Step 3: Substitute for f(2)

To find \( f(2) \), substitute \( x = 2 \) into the function: \( f(2) = \frac{1}{2} \times 2 \). Simplify to get \( f(2) = 1 \).

Step 4: Substitute for f(-2)

To find \( f(-2) \), substitute \( x = -2 \) into the function: \( f(-2) = \frac{1}{2} \times (-2) \). Simplify to get \( f(-2) = -1 \).

Key Concepts

Function Evaluation

Function evaluation is the process by which you determine the value of a function for a specific input. Functions act like machines where you input a number, and it outputs another number according to a given rule. In our example, the function is defined as \[ f(x) = \frac{1}{2}x \] Here, the input is represented by \( x \), and the function outputs the value by multiplying \( x \) by \( \frac{1}{2} \).To evaluate \( f \), simply substitute the specific value of \( x \) into the function and perform the arithmetic operations. For example:

• For \( f(0) \), substitute \( x = 0 \) into the expression, which simplifies to \( 0 \).
• For \( f(2) \), substitute \( x = 2 \), the expression \( \frac{1}{2} \times 2 \) results in \( 1 \).
• Likewise, \( f(-2) \) becomes \( -1 \) when you substitute \( x = -2 \).

Thus, evaluating a function is straightforward once you understand the given rule and apply it to different inputs.

Substitution Method

The substitution method involves replacing the variable in a mathematical expression with a specific value to find the result. This is a fundamental concept in algebra that simplifies finding outputs for functions.When working with the function \( f(x) = \frac{1}{2}x \), you use substitution by replacing \( x \) with given values.For example:

• To find \( f(0) \), substitute \( 0 \) for \( x \), resulting in \( f(0) = \frac{1}{2} \times 0 = 0 \).
• For \( f(2) \), substitute \( 2 \) for \( x \), resulting in \( f(2) = \frac{1}{2} \times 2 = 1 \).
• Lastly, substitute \( -2 \) for \( x \) to find \( f(-2) = \frac{1}{2} \times (-2) = -1 \).

By using substitution, you can easily calculate the value of a function for any given inputs. This method is crucial for understanding and applying function evaluation.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (such as +, -, *, /) that represent a particular quantity. They are essentially instructions for what calculations to perform.In our function example, \( f(x) = \frac{1}{2}x \), the expression "\( \frac{1}{2}x \)" shows how \( x \) is processed to determine the output. Here:

• \( \frac{1}{2} \) is a constant multiplier to the variable \( x \).
• \( x \) is a variable that changes based on input values.

Understanding algebraic expressions is key to solving problems in algebra since they provide the blueprint for calculating function values. By interpreting these expressions correctly, you can substitute the correct numerical values and carry out the calculations with ease.