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Chapter 3: Problem 53

If \(f(x)=3 x+3, g(x)=4 x^{2}-6 x+3,\) and \(h(x)=5 x^{2}-7\) find the following. $$ f(4) $$

Short Answer

Expert verified
\( f(4) = 15 \)

Step by step solution

01

Identify the function

We start with the function given by the problem, which is \( f(x) = 3x + 3 \). This is a linear function where \( 3 \) is the coefficient of \( x \) and the constant is also \( 3 \).
02

Substitute into the function

To find \( f(4) \), substitute \( 4 \) into the function \( f(x) = 3x + 3 \). This gives us the equation: \( f(4) = 3(4) + 3 \).
03

Perform the multiplication and addition

First, perform the multiplication: \( 3(4) = 12 \). Then, add the constant: \( 12 + 3 \).
04

Final calculation

Add the numbers from the previous step: \( 12 + 3 = 15 \). Therefore, \( f(4) = 15 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Substitution
When working with functions, a key technique to master is function substitution. This involves replacing the variable in a function with a given value or another expression. In the context of our exercise, we are given the function \( f(x) = 3x + 3 \) and need to find \( f(4) \). This means substituting \( x \) with \( 4 \) in the function. By doing so, we transform the expression from \( f(x) = 3x + 3 \) to \( f(4) = 3(4) + 3 \).
  • First, locate the variable \( x \) in the function's formula.
  • Then, replace every instance of \( x \) with the number \( 4 \).
  • The resulting expression will then be a straightforward arithmetic calculation.
This process is fundamental for determining the output of functions for specific inputs, enabling us to evaluate how functions behave.
Arithmetic Operations
Arithmetic operations are the basic calculations performed on numbers which include addition, subtraction, multiplication, and division. For evaluating the function substitution \( f(4) = 3(4) + 3 \), we employ multiplication and addition.
  • Multiplication: First, multiply \( 3 \times 4 \). Multiplication indicates how many times one number is added to itself. Here, it means adding \( 3 \) four times, resulting in \( 12 \).
  • Addition: Next, take the product \( 12 \) and add \( 3 \) to it. Addition is combining the two numbers into one total.
So, through these steps, we compute the final result for \( f(4) \), which is \( 15 \). Clear understanding and execution of arithmetic operations allow us to tackle a wide range of mathematical problems with confidence.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain ordinary numbers, variables (like \( x \)), and operators (like add, subtract, multiply, and divide). In our situation, \( 3x + 3 \) is an algebraic expression representing the function \( f(x) \). An essential skill is manipulating these expressions to evaluate functions:
  • An expression like \( 3x + 3 \) consists of the term \( 3x \), which includes a coefficient (\( 3 \)) and a variable (\( x \)), and a constant term (\( 3 \)).
  • When substituting, ensure every instance of the variable is replaced, and then use arithmetic operations to solve.
  • Simplify systematically by following the order of operations: handle multiplication before addition or subtraction.
Understanding algebraic expressions and how to manage them is crucial in studying algebra. It helps in constructing and solving equations, thus opening pathways to comprehend more complex mathematical concepts.

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Most popular questions from this chapter

Given the following functions, find the indicated values. \(h(x)=-x^{2}\) a. \(h(-5)\) b. \(h\left(-\frac{1}{3}\right)\) c. \(h\left(\frac{1}{3}\right)\)

Complete the given table and use the table to graph the linear equation. $$x-y=-5$$ $$\begin{array}{|c|c|c|c|} \hline x & {0} & {} & {1} \\ \hline y & {} & {0} & {} \\ \hline \end{array}$$

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