Problem 31 - Beginning and Intermediate Algebra

Question

Let \(f(x)=3 x-7\) and \(g(x)=x^{2}-4 x-9 .\) Find each of the following and simplify. $$f(a)$$

Step-by-Step Solution

Answer

\(f(a) = 3a - 7\)

Step 1: Substitute a for x in the function f(x)

As given, \(f(x) = 3x - 7\). We are asked to find the value of the function when the input is \(a\). So, we will substitute \(x\) with \(a\) in the function.

Step 2: Evaluate the function f(a)

Now that we have substituted \(a\) in the function, we need to evaluate the function for this new input. The function becomes: \[f(a) = 3a - 7\]

Step 3: Simplify the expression f(a)

The final expression for \(f(a)\) is already in its simplest form, no further simplification needed. \[f(a) = 3a - 7\] So, the value of the function f(x) when x equals to a is \(f(a) = 3a - 7\).

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions. They are characterized by their straight-line graphs on the coordinate plane. A linear function is typically written in the form \( f(x) = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. The slope \( m \) determines the steepness and direction of the line, while the y-intercept \( b \) is the point where the line crosses the y-axis.

In our exercise, the function \( f(x) = 3x - 7 \) is a linear function. Here, the slope \( m \) is 3, meaning the line rises rapidly, moving up 3 units for every 1 unit it moves to the right. The y-intercept \( b \) is -7, indicating that the line crosses the y-axis at the point (0, -7).

Linear functions are easy to work with because of their predictable behavior. Whenever you see an equation of the form \( mx + b \), you can immediately identify it as a linear function and use this form to evaluate or graph the function.

Function Substitution

Function substitution is a method used to evaluate the value of a function given a particular input. It involves replacing the variable in the function's expression with a specified value or another expression. In our exercise, we're substituting 'a' for 'x' in the function \( f(x) = 3x - 7 \).

Here’s how it works step-by-step:

• Identify the variable in the function. In our case, it is \( x \).
• Replace this variable with the given input. For \( f(a) \), \( x \) is replaced with \( a \).
• Simplify the resulting expression, if needed. After substitution, \( f(a) = 3a - 7 \) which is the simplest form.

Function substitution is a useful technique that helps in calculating specific values of functions or in deriving new functions from existing ones. It shows how versatile functions can be by allowing them to adapt to different inputs.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and arithmetic operations. They can be as simple as a single number or involve several operations. Understanding algebraic expressions is essential when handling functions as it helps in manipulation and simplification.

When dealing with functions like \( f(x) = 3x - 7 \), substitute 'a' for 'x' to get the algebraic expression \( 3a - 7 \). Here, \( 3a - 7 \) consists of:

• Coefficient: 3, which is multiplied by the variable \( a \).
• Variable: \( a \), a symbol representing an unknown or any number in a set.
• Constant: -7, a fixed value subtracted from the expression.

To simplify an algebraic expression with single operations, always combine like terms and perform arithmetic operations where possible. However, in our example, \( 3a - 7 \) is already in its simplest form, showing no further simplification is needed.

Algebraic expressions form the backbone of manipulating functions and equations, making them fundamental in various branches of mathematics.