Problem 9 - Algebra A Combined Function

Question

If \(f(x)=x^{2}-6 x+2, g(x)=-2 x\), and \(h(x)=\sqrt{x}\), find each composition. $$ (f \circ g)(2) $$

Step-by-Step Solution

Answer

(f \circ g)(2) = 42

Step 1: Understand Composition

Composition of functions means plugging one function into another. In this problem, we need to find \((f \circ g)(x)\), which is equal to \(f(g(x))\). This means we must first evaluate \(g(x)\) and then plug that result into \(f(x)\).

Step 2: Evaluate g(2)

Substitute \(2\) into \(g(x) = -2x\). Calculate \(g(2) = -2 \times 2 = -4\).

Step 3: Evaluate f(g(2))

Now substitute \(g(2) = -4\) into \(f(x) = x^2 - 6x + 2\). Calculate \(f(-4) = (-4)^2 - 6(-4) + 2\).

Step 4: Simplify f(-4)

Compute each term in the expression: \((-4)^2 = 16\), \(-6(-4) = 24\), and \(+2\). Thus, \(f(-4) = 16 + 24 + 2 = 42\).

Key Concepts

Polynomials

Polynomials are mathematical expressions that sum several terms, with each term consisting of a variable raised to a non-negative integer power. In simple terms, polynomials look like sequences of numbers combined with variables and operations of addition, subtraction, and multiplication. A polynomial can have one or more terms, and each term is a product of a constant (known as a coefficient) and a variable raised to a power.

For example, the function provided in the original exercise is a quadratic polynomial, given by:

• \(f(x) = x^2 - 6x + 2\)

This polynomial has three terms: \(x^2\) (a term with the variable raised to the power of 2), -6x (the variable is raised to the power of 1), and the constant term 2. Whether you are dealing with simple or complex polynomials, understanding the relationship between these terms is crucial for performing mathematical calculations such as addition, subtraction, and evaluating function compositions.

In function composition, you substitute entire expressions for the variable, requiring you to evaluate these polynomials at specific values. As you'll see, evaluating polynomials is a stepping stone in finding the composition of functions, which we'll explore next.

Square Roots

Square roots are fundamentally about finding a number that, when multiplied by itself, equals the original number. They are the inverse of squaring a number. In mathematics, the square root operation is represented by the radical symbol \(\sqrt{}\). For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

In the context of the exercise, we have the function:

• \(h(x) = \sqrt{x}\)

This function transforms each input \(x\) by taking its square root. It's essential when dealing with square root functions to ensure that the value inside the square root is non-negative, as square roots of negative numbers aren't real numbers within standard mathematical systems.

When performing function compositions involving square root functions, it becomes critical to check whether the values substituted make sense within the context of square roots. Square root functions often appear in various calculations due to their unique capability to "reverse" squaring processes.

Linear Functions

Linear functions are among the simplest forms of functions in mathematics. They can be represented with the equation of a line or slope-intercept form: \(y = mx + b\), where "\(m\)" represents the slope and "\(b\)" represents the y-intercept. Linear functions graph as straight lines and have a constant rate of change.

In our original problem, function \(g(x) = -2x\) is a linear function where the slope \(-2\) depicts a line with a downward or negative slope. When substituting values into a linear function, you're evaluating the equation, which results in a single calculated number. This number can then be used in the broader context of function compositions.

Understanding linear functions, particularly when finding compositions like \((f \circ g)(x) = f(g(x))\), is essential. First, substitute the specific value (e.g., 2) into the linear function \(g(x)\), and then take the output from the linear function as input for another function, in this case, a polynomial. This process demonstrates how linear functions often serve as foundational building blocks in mathematical compositions, aligning inputs, and outputs seamlessly.