Problem 59 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

(g \circ f)(-3) = 110.

Step 1: Understand the Problem

We are asked to find \((g \circ f)(-3)\). This means we need to evaluate the composition of functions \(g\) and \(f\) at \(x = -3\).

Step 2: Substitute in the First Function

Start by substituting \(x = -3\) into the function \(f\). This gives us \(f(-3) = 3(-3) - 2\).

Step 3: Calculate the First Function Value

Compute \(f(-3)\). Simplify the expression: \(f(-3) = -9 - 2 = -11\).

Step 4: Substitute into the Second Function

Now, use the result from \(f(-3)\) as the input for \(g(x)\). So, we need to find \(g(-11)\).

Step 5: Evaluate the Second Function

Substitute \(-11\) into \(g(x)\): \(g(-11) = (-11)^2 + (-11)\). Simplify this expression: \(-11^2=121\), so \(g(-11) = 121 - 11 = 110\).

Step 6: Write the Result

The value of the composition \((g \circ f)(-3)\) is 110.

Key Concepts

Evaluate Functions

Evaluating a function basically means finding the output value of the function for a given input value. For instance, when we have a function like \( f(x) = 3x - 2 \), we need to substitute the value of \( x \) into the expression to get \( f(x) \).\
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Let's consider \( x = -3 \) for our function \( f(x) \). We simply replace \( x \) with \( -3 \) in the expression to find \( f(-3) \). The expression becomes \( 3(-3) - 2 \). Then, you perform the arithmetic operations to get the result.\
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• Multiply: \( 3 \times -3 = -9 \)
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• Subtract: \(-9 - 2 = -11 \)
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After evaluating, we find that \( f(-3) = -11 \). This process of evaluating helps in determining what the function does to a particular input.

Composition of Functions

Composition of functions is a way of combining two functions where the output of one function becomes the input of the other. This operation is notated as \( (g \circ f)(x) \), which reads as \( g \text{ of } f \text{ at } x \). It means you'll first find \( f(x) \) and then use this result as the input for \( g(x) \).\
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In simpler terms: \

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• Calculate \( f(x) \) for the given \( x \).
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• Use the result from \( f(x) \) as the input for \( g(x) \).
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Using the problem as an example, for \( x = -3 \), we found that \( f(-3) = -11 \). We then input \(-11\) into \( g(x) = x^2 + x \) to get \( g(-11) \). This step-by-step approach simplifies tackling the complexity that may arise from dealing with multiple functions at once.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators (like \(+\) and \(-\)). These expressions can be evaluated to find their output for specific variable values.\
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In our exercise, the functions \( f(x) = 3x - 2 \) and \( g(x) = x^2 + x \) are both algebraic expressions. They show a relationship between \( x \) and the result of the expression.\
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To manipulate and work with these expressions efficiently, follow these key steps:\

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• Substitute the provided variable value into the expression.
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• Replace the variable with the provided number throughout the expression.
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• Calculate the expression by following the order of operations (parentheses, exponents, multiplication and division, addition and subtraction).
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Understanding algebraic expressions is fundamental when working with functions, as they form the basis of evaluating and composing functions.