Problem 58 - Elementary and Intermediate Algebra

Question

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+x .\) Find each of the following. $$ (g \circ f)(4) $$

Step-by-Step Solution

Answer

\((g \circ f)(4) = 110\).

Step 1: Understand the Composition of Functions

The problem asks us to evaluate the composition of two functions, denoted as \((g \circ f)(4)\). This expression means we need to find \(g(f(4))\), which requires first finding \(f(4)\) and then using that result in the function \(g(x)\).

Step 2: Evaluate f(4)

The function \(f(x) = 3x - 2\). Substitute \(x = 4\) into the function:\[f(4) = 3(4) - 2 = 12 - 2 = 10\]So, \(f(4) = 10\).

Step 3: Substitute f(4) into g(x)

Now that we know \(f(4) = 10\), substitute it into the function \(g(x) = x^2 + x\). Thus, we need to find \(g(10)\):\[g(10) = 10^2 + 10 = 100 + 10 = 110\]

Step 4: Combine Results

The composition of functions results in \((g \circ f)(4) = g(f(4)) = g(10) = 110\).

Key Concepts

Evaluating Functions

Evaluating functions is an essential concept in mathematics that involves finding the output value of a function for a given input. Imagine a function as a machine that takes an input, performs a specific operation, and then spits out an output. For example, if we have a function defined as \( f(x) = 3x - 2 \), and we want to evaluate it at \( x = 4 \), we substitute 4 in place of \( x \) in the expression:

• First, replace \( x \) with 4: \( f(4) = 3(4) - 2 \)
• Calculate the expression: \( f(4) = 12 - 2 = 10 \)

So, the output of the function \( f \) at \( x = 4 \) is 10. It’s important to follow this procedure step by step to ensure accuracy. Evaluating functions forms the foundation of more complex operations like function composition.

Substitution in Algebra

Substitution in algebra is a technique used to solve equations or evaluate expressions by replacing variables with numbers. It is a straightforward process but requires careful calculation. When you have evaluated \( f(4) \) to be 10 in a problem involving the composition of functions, the next step is using substitution to find the value of another function \( g(x) \), using the result from \( f(x) \):

• Given \( g(x) = x^2 + x \), substitute 10 for \( x \): \( g(10) = 10^2 + 10 \)
• Simplify the equation step by step: \( g(10) = 100 + 10 = 110 \)

Substitution makes solving and evaluating functions more manageable, as it allows you to focus on one part of the problem at a time. By substituting and then simplifying, you can effectively handle even seemingly complex algebraic expressions.

Algebraic Functions

Algebraic functions are expressions involving one or more variables and arithmetic operations like addition, subtraction, multiplication, division, or exponentiation. These functions are common in mathematics and are essential for understanding various calculus concepts. When working with algebraic functions like \( f(x) = 3x - 2 \) and \( g(x) = x^2 + x \), one fundamental operation is function composition.Function composition involves plugging the output of one function into another, written as \((g \circ f)(x) = g(f(x))\). In our example, to find \((g \circ f)(4)\), we first calculate \( f(4) \) to get the result 10, then substitute this into the function \( g \) to find \( g(10) = 110 \).This process showcases how algebraic functions can be combined to produce new functions, illustrating the interconnectedness of algebraic concepts. Understanding these relationships deepens your comprehension of how functions work and prepares you for more advanced studies in mathematics.