Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Let \(f(x)=x^{2}\) and \(g(x)=2 x-5 .\) Match each function in Group I with the correct expression in Group $I$$$I$$$$(g \circ f)(x)$$ $$\mathbf{II}$$$$A.\quad4 x^{2}-20 x+25$$$$B.\quad x^{2}-2 x+5$$$$C.\quad 2 x^{2}-5$$$$D.\quad \frac{x^{2}}{2 x-5}$$$$E. \quad x^{2}+2 x-5$$$$F. \quad 2 x^{3}-5 x^{2}$$

Step-by-Step Solution

Answer

The correct match is (g \circ f)(x) = C. 2x^2 - 5.

Step 1: Understand the Composition of Functions

The composition of two functions, \((g \circ f)(x)\), means we first apply the function \(f(x)\) and then apply function \(g\) to the result. This can be expressed as \(g(f(x))\).

Step 2: Substitute f(x) into g(x)

Given \(f(x) = x^2\) and \(g(x) = 2x - 5\), substitute \(x^2\) for \(x\) in the function \(g(x)\). Thus, \(g(f(x)) = g(x^2) = 2(x^2) - 5\).

Step 3: Simplify the Expression

Simplifying the expression \(2(x^2) - 5\) results in \(2x^2 - 5\). This is the expression for \((g \circ f)(x)\).

Step 4: Match the Expression

Compare \(2x^2 - 5\) with the given options in Group II: \(C.\) matches the expression \(2x^2 - 5\).

Key Concepts

Function Notation

Function notation is a way to represent functions in a clear and concise manner. It typically uses the format of a letter followed by a variable in parentheses, like \(f(x)\). This notation helps us understand which function we are dealing with and what the input variable is.
Function notation specifies how a function, represented as \(f\), maps an input \(x\) to an output \(f(x)\). In our exercise, you encounter functions such as \(f(x) = x^2\) and \(g(x) = 2x - 5\). Here, \(f\) and \(g\) are the function names, while \(x\) is the independent variable. The expressions \(x^2\) and \(2x - 5\) are the rules defining these functions.

• Function notation is useful for expressing complex operations succinctly, such as function composition.
• It simplifies tracking operations when dealing with multiple functions.
• The notation quickly conveys the relationship between input and output values.

Understanding function notation is foundational when working with compositions, transformations, or evaluations of functions.

Quadratic Functions

Quadratic functions are a type of polynomial function where the highest power of the variable is squared. They are represented generally as \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero.
In our exercise, the function \(f(x) = x^2\) is a simple quadratic function where \(a=1\), \(b=0\), and \(c=0\). It illustrates the basic form of a quadratic function.

• Quadratic functions produce parabolic graphs open either upwards or downwards based on the sign of \(a\).
• These functions often model various real-world phenomena, such as projectile motion.
• The vertex form and factored form are other common ways to express quadratic functions for different analytical purposes.

Quadratic functions are crucial in mathematics and are frequently oberserved in the context of solving equations or understanding dynamic systems.

Function Operations

Function operations involve the various ways you can manipulate functions, such as adding, subtracting, multiplying, dividing, or composing them. In the example exercise, the focus is on the composition of two functions, \(f(x)\) and \(g(x)\).
Composition, noted as \((g \circ f)(x)\), implies you apply one function first and then the other. It is calculated here as \(g(f(x))\), meaning you input result of \(f(x)\) into \(g(x)\). For our example, after substituting \(f(x) = x^2\) into \(g(x) = 2x - 5\), you get \(2(x^2) - 5\). This simplifies to \(2x^2 - 5\).

• Understanding function operations allows combining functions to form new ones with desirable properties.
• Aids in the analysis of how changes in one input can propagate through a chain of functions.
• Essential for problem-solving in areas such as calculus and beyond.

Mastering function operations enhances mathematical fluency and problem-solving skills.