Problem 89 - 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. $$ f(4) $$

Step-by-Step Solution

Answer

\(f(4) = 10\)

Step 1 - Identify the function for the given value

The problem asks to evaluate the function \(f(x)\) for \(x = 4\). The function \(f(x)\) is given as \(f(x) = 3x - 2\).

Step 2 - Substitute the given value of x into the function

Substitute \(x = 4\) into the function \(f(x)\). This gives us \(f(4) = 3(4) - 2\).

Step 3 - Calculate the result

Perform the multiplication and subtraction: \(f(4) = 3 \times 4 - 2 = 12 - 2\).

Step 4 - Simplify the expression

Simplify the expression to get the final result: \(12 - 2 = 10\).

Key Concepts

algebraic functions

Algebraic functions are mathematical expressions involving variables and constants. An algebraic function like \( f(x) = 3x - 2 \) consists of coefficients (3), variables (x), and constants (-2). These elements combine through basic arithmetic operations such as addition, subtraction, multiplication, and division.
Understanding how to work with algebraic functions is essential because it forms the basis of more complex mathematical concepts. When you see a function like \( f(x) \), it means that for any given value of \( x \), you can calculate \( f(x) \) based on the formula provided.
In the given exercise, the function \( f(x) = 3x - 2 \) shows a linear relationship between \( x \) and \( f(x) \). Want to visualize it? Think of it as a straight line you could plot on a graph.

substitution in functions

Substitution in functions is a critical step in evaluating algebraic functions. When asked to find \( f(4) \), you need to substitute the value of 4 wherever you see \( x \) in the function.
This concept can be outlined as follows:

• Identify the function. For example, \( f(x) = 3x - 2 \).
• Recognize the input value that you need to substitute for \( x \). Here, it's \( x = 4 \).
• Replace \( x \) with the given value. So, \( f(4) = 3(4) - 2 \).

Substitution helps simplify the function and make it easier to solve. It's like replacing a placeholder with an actual number to find the result.
Understanding this concept aids in making accurate problem-solving steps, reducing errors, and ensuring the correct application of the function.

simplifying expressions

Simplifying expressions involves performing arithmetic operations to reduce them to their simplest form. After substitution, you'll often need to simplify to find the final result.
In the given problem, the function after substitution is \( f(4) = 3(4) - 2 \). Here's how you simplify this:

• First, perform the multiplication: \( 3 \times 4 = 12 \).
• Then, carry out the subtraction: \( 12 - 2 = 10 \).

By simplifying, you ensure the expression is manageable and the result is easy to interpret. Proper simplification techniques make complex problems more straightforward.
Always remember to follow the order of operations, which is multiplication and division first, then addition and subtraction.

step-by-step solution

A step-by-step solution helps break down the problem into manageable parts. This method is particularly useful in understanding and solving algebraic functions.
Here's a recap of the steps to solve \( f(4) \) for the given function \( f(x) = 3x - 2 \):
Step 1: Identify the function and the value to be substituted. Here, \( f(x) \)=3x - 2 and \( x = 4 \).
Step 2: Substitute the given value into the function: \( f(4) = 3(4) - 2 \).
Step 3: Perform the arithmetic operations. First, multiply to get 12. Then, subtract 2 from 12.
Step 4: Simplify the result. The final answer is \( f(4) = 10 \).
Following these steps ensures accuracy and clarity in solving functions. It's like solving a puzzle – each piece (or step) brings you closer to the final picture (or solution).
Remember, practice makes perfect, and using step-by-step solutions can make even the most challenging problems easier to tackle.