Problem 42 - Beginning and Intermediate Algebra

Question

Let \(f(x)=-5 x+2\) and \(g(x)=x^{2}+7 x+2 .\) Find each of the following and simplify. $$f(7)$$

Step-by-Step Solution

Answer

The short answer is: \(f(7) = -33\).

Step 1: Write down the given function for \(f(x)\)

We are given that \(f(x) = -5x + 2\).

Step 2: Substitute \(x = 7\) into the function

To find \(f(7)\), replace \(x\) with \(7\) in the equation for \(f(x)\): \(f(7) = -5(7) + 2\)

Step 3: Evaluate and simplify

Now simplify the expression by performing the operations: \(f(7) = -35 + 2 = -33\) So, the value of \(f(7)\) is \(-33\).

Key Concepts

Linear Functions

Linear functions are a type of mathematical function where each function is represented by a straight line when you graph it on a coordinate plane. The general form of a linear function is often written as \(f(x) = mx + b\), where \(m\) represents the slope of the line, and \(b\) indicates the y-intercept.
When you look at the function \(f(x) = -5x + 2\), you see that it fits this standard format. Here, the slope \(m = -5\) shows how steep the line is, and the y-intercept \(b = 2\) is where the line crosses the y-axis.
This means, for every increase of 1 in \(x\), the function decreases by 5. Linear functions are straightforward and a fundamental part of algebra, making them powerful tools for various analyses and predictions.

• Recognizing the formula: \(f(x) = mx + b\)
• Understanding the role of the slope and y-intercept
• Ability to predict changes using the slope

Algebraic Substitution

Algebraic substitution is a method used in algebra to simplify or solve equations by replacing variables with given numbers or other expressions. In our exercise, we are asked to find \(f(7)\) using the function \(f(x) = -5x + 2\). This requires substituting the value of 7 for each occurrence of \(x\) in the function.
Here's how you do it:

• Identify what's given: you have a function \(f(x) = -5x + 2\).
• Replace \(x\) with 7: plug in the value directly where \(x\) appears.
• The expression becomes \(-5(7) + 2\).

Substitution helps in transitioning from a variable-dependent form to a numerical form, which makes it easier to evaluate or simplify further.
It is a simple and powerful technique commonly used across various math problems to simplify expressions and reach solutions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves performing operations to reduce an expression to its simplest form, thereby making it clearer and easier to use. After substituting 7 in place of \(x\) in the function \(f(x) = -5x + 2\), we got the expression \(-5(7) + 2\).
Let's simplify it step-by-step:

• First, multiply: Calculate \(-5 \times 7\) to get \(-35\).
• Next, add: Take \(-35 + 2\).
• Finally, simplify: This results in \(-33\).

Simplifying helps others understand your math work and ensures your final answer is as clear as possible.
In general, always follow the order of operations (PEMDAS/BODMAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to ensure accurate simplification.