Problem 49 - Elementary and Intermediate Algebra

Question

Find \(f(3)\) and \(f(-1) .\) See Example 4. $$ f(x)=2 x-3 $$

Step-by-Step Solution

Answer

\( f(3) = 3 \) and \( f(-1) = -5 \).

Step 1: Understand the Function

The function provided is a simple linear function: \( f(x) = 2x - 3 \). To find \( f(3) \) and \( f(-1) \), we need to substitute these values into the function.

Step 2: Finding \( f(3) \)

Substitute \( x = 3 \) into the function.\[ f(3) = 2(3) - 3\]Now, carry out the multiplication and subtraction:\[ f(3) = 6 - 3\]Finally, calculate the result:\[ f(3) = 3\]

Step 3: Finding \( f(-1) \)

Substitute \( x = -1 \) into the function.\[ f(-1) = 2(-1) - 3\]Perform the multiplication:\[ f(-1) = -2 - 3\]Then, evaluate the subtraction:\[ f(-1) = -5\]

Key Concepts

Function Evaluation

When we talk about function evaluation, we're discussing how to find the output of a function for a specific input value. In the context of linear functions, which are defined by expressions like \( f(x) = 2x - 3 \), evaluating the function means plugging in the given values of \( x \) to see what the corresponding output or \( f(x) \) will be.
Function evaluation can be understood as a two-step process:

• Substitute the specific value of \( x \) into the function.
• Perform the necessary arithmetic operations to simplify and solve the expression.

For example, to evaluate the function at \( x = 3 \), you substitute \( 3 \) for \( x \), resulting in the calculations \( f(3) = 2(3) - 3 \) which simplifies to \( 3 \). Function evaluation is a fundamental skill that helps us understand how functions behave and what outputs we can expect for different inputs.

Substitution Method

The substitution method is a straightforward approach used in algebra to find the value of a function. It's especially handy when working with expressions and equations.
Substitution involves:

• Replacing a variable in a mathematical expression with a given number.
• Performing the necessary calculations to find the result.

Let's see how it works with an example:
Given the function \( f(x) = 2x - 3 \), and asked to find \( f(3) \), we substitute \( x = 3 \) into the function equation. This means we replace every \( x \) with \( 3 \). It becomes:

• \( f(3) = 2 \times 3 - 3 \).
• \( f(3) = 6 - 3 \).
• \( f(3) = 3 \).

This method allows you to evaluate functions quickly by methodically following the steps of substitution and calculation.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. When looking at linear functions like \( f(x) = 2x - 3 \), we have a simple algebraic expression where \( x \) is the variable and 2 and -3 are constants.
Algebraic expressions can help describe relationships and patterns using variables, making them integral to solving equations and analyzing functions.
When working with algebraic expressions, remember:

• Understand the operations: In \( f(x) = 2x - 3 \), \( 2x \) indicates multiplication, and \(-3 \) indicates subtraction.
• Apply the order of operations: When evaluating, perform multiplication and division before addition and subtraction.

Through algebraic expressions, we can model real-world situations, explore relationships, and compute function values like \( f(3) \) and \( f(-1) \). These expressions are foundational tools in mathematics, allowing us to convey complex concepts in a structured form.