Problem 67 - A Graphical Approach to Precalculus with Limits

Question

For the given \(f(x)\) find (a) \(f(a),\) (b) \(f(b+1),\) and (c) \(f(3 x)\). $$f(x)=2 x-5$$

Step-by-Step Solution

Answer

(a) \(f(a) = 2a - 5\), (b) \(f(b+1) = 2b - 3\), (c) \(f(3x) = 6x - 5\).

Step 1: Substitute a in the Function

To find \(f(a)\), substitute \(a\) in place of \(x\) in the function. The function is \(f(x) = 2x - 5\), so \(f(a) = 2a - 5\).

Step 2: Substitute b+1 in the Function

To find \(f(b+1)\), substitute \(b+1\) in place of \(x\) in the function. Replace \(x\) with \(b+1\) in \(f(x) = 2x - 5\), giving the expression \(f(b+1) = 2(b+1) - 5\), which simplifies to \(2b + 2 - 5 = 2b - 3\).

Step 3: Substitute 3x in the Function

To find \(f(3x)\), substitute \(3x\) in place of \(x\) in the function. Replace \(x\) with \(3x\) in \(f(x) = 2x - 5\), resulting in \(f(3x) = 2(3x) - 5\), which simplifies to \(6x - 5\).

Key Concepts

Substitution

Substitution is a fundamental technique in algebra that involves replacing a variable with a specific value or expression. It's a way to evaluate functions for specific inputs. In the context of our example, the function \( f(x) = 2x - 5 \), we substitute different expressions like \( a \), \( b+1 \), and \( 3x \) into \( x \) to evaluate the function at these particular points.
When performing substitution, follow these easy steps:

• Identify the variable in the function that needs replacement, which is \( x \) in this case.
• Substitute the given value or expression wherever \( x \) appears in the function.
• Simplify the resulting expression if necessary.

Substitution is an essential skill to solve different kinds of mathematical problems, and it's especially useful for function evaluation. It's like putting different ingredients into a recipe to see what comes out. Practice this with various values, and soon you'll be substituting effortlessly!

Linear Functions

Linear functions are mathematical expressions that create straight lines when graphed. These functions have the standard form of \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In our exercise, the function \( f(x) = 2x - 5 \) is linear.
Key characteristics of linear functions include:

• The graph is a straight line.
• The slope \( m \) determines the angle of the line. Positive slopes rise, negative slopes fall.
• The y-intercept \( b \) indicates where the line crosses the y-axis.

Linear functions are simple yet powerful in modeling relationships in many real-life situations, such as calculating taxes or converting units. Understanding these concepts helps in various applications both in mathematics and beyond.

Simplifying Expressions

Simplifying expressions means making them easier to understand and work with by combining like terms and applying arithmetic operations. After substitution in our function \( f(x) = 2x - 5 \), simplifying is often required to achieve a clean result.
Here's how you can simplify expressions effectively:

• Distribute any coefficients to terms inside parentheses.
• Combine like terms, which are terms containing the same variable to the same power.
• Perform basic arithmetic operations like addition and subtraction to reduce the expression.

For example, in the solution \( f(b+1) = 2(b+1) - 5 \), simplify by distributing the \( 2 \) to both \( b \) and \( 1 \), resulting in \( 2b + 2 - 5 \), and then combine to get \( 2b - 3 \). By simplifying, you make expressions easier to interpret and solve, which is a valuable technique in mathematics.