Problem 32 - Algebra 2

Question

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (f \circ g)(-2) $$

Step-by-Step Solution

Answer

The value of \((f \circ g)(-2)\) is 25.

Step 1: Understand the Functions

The given functions are \(f(x)=x^{2}\) and \(g(x)=x-3\). These are straightforward polynomial functions.

Step 2: Substitute g into f

We need to find \((f \circ g)(-2)\), which means plugging g(-2) into the f function. First, calculate \(g(-2)\): Just substitute -2 into \(g(x)\), which gives us \(g(-2)=-2-3=-5\).

Step 3: Compute the Result

Now that we have found g(-2) to be -5, we substitute it into the f function. That gives us \(f(g(-2))=f(-5)=(-5)^{2}=25\).

Key Concepts

Polynomial Functions

Polynomial functions are a type of mathematical expression that involve variables raised to whole-number exponents and coefficients that are real numbers. They are essential in various areas of math because they offer simple models expressing real-world curves and behaviors. A polynomial function is usually in the form of:

• A constant term which is a number without a variable (e.g., 3).
• Linear terms, which are variables with an exponent of one (e.g., \(2x\)).
• Quadratic terms, which are variables squared (e.g., \(x^2\)).
• Cubic terms, which have a variable raised to the third power (e.g., \(x^3\)).

Consider the polynomial function \(f(x) = x^2\). It is a quadratic function, meaning its highest exponent is 2, forming a parabola when graphed. In the given exercise, both functions \(f(x) = x^2\) and \(g(x) = x - 3\) are polynomial functions, with \(f(x)\) being quadratic and \(g(x)\) being linear.

Composite Functions

Composite functions involve combining two functions, where the output of one function becomes the input of another. To denote a composite function, we use the symbol \(\circ\), read as "of" or "compose". Thus,

• \((f \circ g)(x)\) means the function \(g(x)\) is inside the function \(f(x)\).
• In practical terms, you first apply \(g(x)\) and then use its result as the argument for \(f(x)\).

For instance, in our given problem of \((f \circ g)(-2)\), we start by calculating \(g(-2)\), and then take this value and substitute it into \(f\). This creates a new composite function based on two simpler functions. Exploring composition helps deepen your understanding of how functions interact, leading to more complex expressions.

Function Evaluation

Function evaluation is the process of determining the output of a function given an input. It simply involves substituting a specific value into a function. This is crucial because it allows us to quantify and analyze relationships and changes between variables. Let's break it down:

• To evaluate \(g(x)\) at \(x = -2\), we replace \(x\) with \(-2\), which results in: \[g(-2) = -2 - 3 = -5\].
• Next, we take \(g(-2) = -5\) and substitute it into \(f(x)\), giving us: \[f(g(-2)) = f(-5) = (-5)^2 = 25\].

The purpose of this evaluation process is to find specific values of composite functions accurately. By evaluating correctly, students can check their understanding and ensure their ability to perform operations on functions effectively.