Problem 90 - Algebra for College Students

Question

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ f(100) $$

Step-by-Step Solution

Answer

298

Step 1: Understand the Function

Identify the function that needs to be evaluated. The function given is \(f(x) = 3x - 2\).

Step 2: Substitute the Value

Substitute \(x = 100\) into the function \(f(x)\). This means we replace \(x\) with 100 in the expression \(3x - 2\).

Step 3: Perform the Calculation

Calculate the value by performing the arithmetic operations:\(f(100) = 3(100) - 2\)

Step 4: Simplify the Expression

Simplify the expression from the previous step:\(f(100) = 300 - 2\)Therefore, \(f(100) = 298\).

Key Concepts

Substitution

Substitution is the process of replacing a variable with a given value. In our example, we started with the function \( f(x) = 3x - 2 \). Our goal was to find the value of the function when \( x = 100 \). To do this, we replaced the variable \( x \) with 100.
Substitution is a key step in evaluating functions. It transforms the function from a general form into a specific computation. Think of it as if you are translating a formula into a real-world scenario by inputting actual numbers instead of variables.
Here’s how it looks:
- Original function: \(f(x) = 3x - 2\)
- Substitute \(x = 100\)
- New expression: \(f(100) = 3(100) - 2\)
Once we substitute the value for the variable, it’s time to move on to the arithmetic operations.

Arithmetic Operations

Arithmetic operations are basic mathematical calculations that include addition, subtraction, multiplication, and division. These are the steps we need to follow to evaluate the function once we've substituted the variable.
In our example, after substitution, we get the expression \( 3(100) - 2 \).
Here are the detailed operations:
- First, perform the multiplication: \( 3 \times 100 = 300 \)
- Next, perform the subtraction: \( 300 - 2 \)
Breaking down the expression into smaller parts makes it easier to handle and avoid mistakes. This step-by-step approach can be helpful when working with more complicated functions or multiple variables.
Arithmetic operations follow specific rules, often referred to as the order of operations, which can be remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). Understanding these rules ensures you carry out the calculations correctly.

Simplification

Simplification is the process of making an expression easier to understand or solve. After performing the required arithmetic operations, you'll often be left with an expression that needs to be simplified.
In our example, after substituting and performing the arithmetic operations, we got \( 300 - 2 \).
Simplify this to get the final answer:
- \( 300 - 2 = 298 \)
Simplification helps in presenting the solution in its easiest form. It reduces complexity and confusion.
Even though the expression \( 300 - 2 \) is pretty straightforward, the concept of simplification can involve more complex steps for different problems. It's about making sure your final answer is as simple and clear as possible.
By following these steps of substitution, arithmetic operations, and simplification, we found that \( f(100) = 298 \). This systematic approach can be applied to any function evaluation problem, ensuring clarity and accuracy in your solutions.