Problem 62 - Elementary and Intermediate Algebra

Question

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+x .\) Find each of the following. $$ (f \circ g)(0) $$

Step-by-Step Solution

Answer

The value of \((f \circ g)(0)\) is \(-2\).

Step 1: Understand the Composition of Functions

Function composition involves plugging one function into another. \((f \circ g)(x)\) is the same as \(f(g(x))\), meaning we first apply \(g(x)\), and then apply \(f\) to the result.

Step 2: Evaluate \(g(0)\)

The function \(g(x) = x^2 + x\). To find \(g(0)\), substitute \(x = 0\) into the equation: \[ g(0) = 0^2 + 0 = 0. \]

Step 3: Evaluate \(f(g(0))\)

Now that we know \(g(0) = 0\), we evaluate \(f(0)\) where \(f(x) = 3x - 2\). Substitute \(x = 0\) into \(f(x)\): \[ f(0) = 3 \times 0 - 2 = -2. \]

Step 4: Conclusion for \((f \circ g)(0)\)

Since \(f(g(0)) = f(0) = -2\), the result of the composition \((f \circ g)(0)\) is \(-2\).

Key Concepts

Evaluation of Functions

Evaluating functions involves calculating the result of a function for a specific input. Imagine a function as a machine that takes an input, does something to it, and spits out an output. In our example, the functions are given by:

• \(f(x) = 3x - 2\)
• \(g(x) = x^2 + x\)

To evaluate \(g(0)\): - Substitute \(0\) for \(x\) in \(g(x)\). - So, \(g(0) = 0^2 + 0 = 0\). Once we have \(g(0)\), we can use this result to evaluate \(f(x)\). Substitute the output of \(g(x)\), which is \(0\), into \(f(x)\): - \(f(0) = 3 \times 0 - 2 = -2\). Evaluating functions is all about substituting and simplifying to find your final answer.

Mathematical Operations

In the context of function evaluation, basic mathematical operations are essential. They include addition, subtraction, multiplication, and exponentiation. Let's go through some of the operations used in our functions:

• In \(g(x) = x^2 + x\), you see exponentiation \(x^2\), which means multiplying \(x\) by itself.
• You also have addition, where \(x\) and \(x^2\) are added together.

When evaluating \(f(x) = 3x - 2\), you apply: - Multiplication to the term \(3x\), multiplying \(3\) by \(x\). - Subtraction is then used to adjust the product by subtracting \(2\). These simple operations are combined and sequenced to yield our final result, \(-2\) when evaluating \(f(g(0))\). Understanding these operations helps in simplifying and solving various algebraic problems.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. In our problem, we have two core expressions: \(f(x) = 3x - 2\) and \(g(x) = x^2 + x\). Understanding how to manipulate these expressions is crucial.

• The expression \(g(x) = x^2 + x\) involves variables and constants combined with addition.
• \(f(x) = 3x - 2\) incorporates multiplication and subtraction.

When dealing with these expressions, it's important to perform operations in the correct order, known as the "order of operations" or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). For instance, when \(x = 0\) in \(g(x)\), calculate the exponent and then perform addition to get \(0\). This result then becomes the input for \(f(x)\). The effective manipulation and evaluation of algebraic expressions are significant skills in algebra, allowing us to solve and comprehend complex mathematical problems.