Problem 47 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

\( f(4) = 15 \)

Step 1: Identify the Function for Evaluation

We need to find the value of the function \( f(x) \) at \( x = 4 \). The given function is \( f(x) = 3x + 3 \).

Step 2: Substitute the Value into the Function

Replace \( x \) with 4 in the function: \[ f(4) = 3(4) + 3 \].

Step 3: Perform the Multiplication

Calculate the multiplication in the expression: \[ 3(4) = 12 \].

Step 4: Add to Complete the Evaluation

Add 3 to the result of the multiplication to find \( f(4) \): \[ 12 + 3 = 15 \].

Step 5: Conclusion: Solution to the Function Value

Thus, \( f(4) = 15 \).

Key Concepts

Algebraic Functions

Algebraic functions are a type of mathematical expression composed of constants and variables in various forms, typically involving addition, subtraction, multiplication, division, and exponentiation. An algebraic function is defined over a set range and domain, and it maps every element from the domain to an element in the range. In our original exercise, we encountered the algebraic function \(f(x) = 3x + 3\), which is a linear function because it represents a line in a two-dimensional plane. Understanding algebraic functions is crucial as they serve as foundational tools in various areas of mathematics and real-world applications, such as calculating interest, determining population growth, or predicting trends.

Substitution in Algebra

Substitution in algebra involves replacing a variable with a specific number or expression. This technique is often used to evaluate functions, simplify expressions, or solve equations. In the step-by-step solution provided, you can see substitution in action when \( x \) was replaced with 4 in the expression \(f(x) = 3x + 3\). By substituting \(x\) with 4, the task shifted to evaluating the numerical outcome of \(f(4) = 3(4) + 3\). This simple yet powerful method allows one to transform an abstract problem into a concrete number, simplifying the process of problem-solving and making algebra more approachable.

Mathematical Operations

Mathematical operations are the processes through which numbers and expressions are combined to form new numbers and expressions. The four primary operations are addition, subtraction, multiplication, and division. In our example, after substituting \(x\) with 4, we performed several mathematical operations to simplify the expression. First, we executed multiplication: \(3(4) = 12\). Following that, an addition operation was completed: \(12 + 3 = 15\). Understanding these basic mathematical operations is essential for manipulating algebraic expressions efficiently and accurately, enabling more straightforward and clear solutions to problems.

Evaluating Linear Equations

Evaluating linear equations is a central skill in algebra. It involves determining the output value of a linear equation given an input value. In simple terms, if you have an equation in the form \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept, you can find \(y\) by substituting the \(x\) value and performing necessary computations. The exercise demonstrated evaluation by calculating \(f(4) = 15\) for the linear function \(f(x) = 3x + 3\). Mastery of evaluating linear equations helps in various academic fields, from economics and physics to computer science, providing solutions to predict, calculate, and interpret data.