Problem 6 - A Graphical Approach to College Algebra

Question

Let \(f(x)=x^{2}\) and \(g(x)=2 x-5 .\) Match each function in Group I with the correct expression in Group II. (Group II) A. \(4 x^{2}-20 x+25\) B. \(x^{2}-2 x+5\) C. \(2 x^{2}-5\) D. \(\frac{x^{2}}{2 x-5}\) E. \(x^{2}+2 x-5\) F. \(2 x^{3}-5 x^{2}\) (Group I) $$(g \circ f)(x)$$

Step-by-Step Solution

Answer

(g \circ f)(x) corresponds to C. 2x^2 - 5.

Step 1: Understanding the Notation

The notation \((g \circ f)(x)\) means the composition of function \(g\) and function \(f\). Specifically, this means applying \(f(x)\) first, and then using the result as the input of \(g(x)\). Mathematically, this can be expressed as \(g(f(x))\).

Step 2: Determine f(x)

Let us find \(f(x)\) for the function given in Group I. We have \(f(x) = x^2\). This is simply the original given \(f(x)\) function.

Step 3: Substitute into g(x)

Next, substitute \(f(x) = x^2\) into \(g(x) = 2x - 5\). This gives us \(g(f(x)) = g(x^2) = 2(x^2) - 5 = 2x^2 - 5\).

Step 4: Match with Group II

We have found that \(g(f(x)) = 2x^2 - 5\). Now, we need to match this result with one of the expressions in Group II. The matching expression is C. \(2x^2 - 5\).

Key Concepts

Function Composition

Function composition is a way of combining two functions by using the output of one function as the input of another. When we talk about the composition of functions, such as \(g \circ f\), we mean that the function \(f(x)\) is applied first, and its output becomes the input for the function \(g(x)\).
For example:

• Given \(f(x) = x^2\) and \(g(x) = 2x - 5\), the composite function \(g \circ f(x)\) is calculated by first applying \(f(x)\).
• We find \(f(x)\) is \(x^2\). Next, use this \(x^2\) as input for \(g(x)\).
• This results in \(g(f(x)) = g(x^2) = 2x^2 - 5\).

Notice that when composing functions, the order in which they are applied matters. \(g \circ f(x)\) is not the same as \(f \circ g(x)\). Each sequence could potentially yield different outputs, so paying attention to the notation is crucial.

Quadratic Functions

Quadratic functions are polynomial functions of degree two and are typically written in the form \(f(x) = ax^2 + bx + c\). These functions graph as parabolas.
In our example, \(f(x) = x^2\) is a simple quadratic function where

• The coefficient of \(x^2\) is \(1\), which means the parabola opens upwards.
• The \(b\) and \(c\) components are \(0\), so its vertex is at the origin (0,0).

Quadratic functions are fundamental in algebra and appear frequently in various contexts, such as physics for motion equations, and economics for profit functions.
Since the key feature of quadratics is their parabola shape, knowing how to compose them with linear functions, like our \(g(x) = 2x - 5\), lets us explore how these combinations can transform into complex expressions, like \(2x^2 - 5\) in our composite function example.

Function Notation

Function notation is a way to mathematically express operations on functions succinctly and clearly. It uses symbols like \(f(x)\) instead of words, telling us exactly what the function does to the input variable, \(x\).
In function notation:

• Top-level always labels the function: \(f\), \(g\), etc.
• The variable \(x\) inside tells what inputs the function takes.

For example, \(f(x) = x^2\) tells us \(f\) is a function squaring any input \(x\). Likewise, \(g(x) = 2x - 5\) expresses a linear function that scales \(x\) twice and subtracts \(5\).
This notation is flexible and universal, providing consistency in mathematics. Not only does it aid in writing expressions clearly, but it also helps in understanding operations like function composition. Function notation thus supports learners in visualizing and performing mathematical operations effectively.