Problem 91 - Algebra 2

Question

If \(f(x)=x^{2}\) and \(g(x)=x-1,\) which statement is true? A. \((f \circ g)(x) \geq(g \circ f)(x)\) for all values of \(x .\) B. \((f \circ g)(x) \leq(g \circ f)(x)\) for all values of \(x\) . C. \((f \circ g)(x)=(g \circ f)(x)\) only for \(x=1\) D. \((f \circ g)(x) \neq(g \circ f)(x)\) for any value of \(x .\)

Step-by-Step Solution

Answer

B. \((f \circ g)(x) \leq(g \circ f)(x)\) for all values of \(x\) .

Step 1: Compute (f ∘ g)(x)

To find the result of (f ∘ g)(x), input g(x) into f(x). Given that \(f(x) = x^2\) and \(g(x) = x - 1\), computing the output of f composed with g gives \(f(g(x)) = (x - 1)^2\)

Step 2: Compute (g ∘ f)(x)

To find the result of (g ∘ f)(x), input f(x) into g(x). Given that \(f(x) = x^2\) and \(g(x) = x - 1\), computing the output of g composed with f gives \(g(f(x)) = x^2 - 1\)

Step 3: Compare (f ∘ g)(x) and (g ∘ f)(x)

Expanding both function results, (f ∘ g)(x) = \(x^2 - 2x + 1\) and (g ∘ f)(x) = \(x^2 - 1\) . Substracting (g ∘ f)(x) from (f ∘ g)(x) gives \( -2x + 2 \). Since this is always negative for any real x value, thus (f ∘ g)(x) <= (g ∘ f)(x) for all values of x.

Key Concepts

Quadratic Functions

Quadratic functions are a type of function where the highest power of the variable is 2, typically taking the form of \( f(x) = ax^2 + bx + c \). These functions generate a parabolic curve when graphed. In the exercise, the function \( f(x) = x^2 \) is a simple quadratic function where the coefficients \( a = 1 \), \( b = 0 \), and \( c = 0 \). Understanding quadratic functions involves recognizing their key characteristics:

• **Vertex**: The turning point of the parabola. For \( f(x) = x^2 \), the vertex is at the origin (0,0).
• **Axis of Symmetry**: The vertical line that divides the parabola into two mirror images. In \( f(x) = x^2 \), this line is \( x = 0 \).
• **Direction of Opening**: The parabola opens upwards since the coefficient \( a = 1 \) is positive.

Acknowledging these aspects helps in visualizing how the function behaves and interacts with other functions in expressions like function composition.

Inequalities in Functions

Inequalities in functions involve comparing two expressions to determine their relative sizes. In this exercise, you are tasked with deciding whether \((f \circ g)(x) \leq (g \circ f)(x)\) holds for all \(x\). These inequalities reveal the comparative behavior of function compositions:

• **Calculate Difference**: After obtaining \((f \circ g)(x) = (x-1)^2\) and \((g \circ f)(x) = x^2 - 1\), we examine \((f \circ g)(x) - (g \circ f)(x)\) which results in \(-2x + 2\).
• **Solving for All \(x\)**: By checking the condition \(-2x + 2 \leq 0\), we find that this is true as \(x\) grows positively or negatively large, proving \((f \circ g)(x) \leq (g \circ f)(x)\).

Inequalities are crucial as they indicate how functions increase, decrease, or neutralize their effects under specific compositions or constraints.

Function Notation

Function notation is a systematic way to express functions using symbols. In mathematics, it helps in easily communicating operations done on functions. For instance, \( f(x) \) represents a function \( f \) applied to the variable \( x \). Let's break down this exercise:

• **Composition Notation**: The symbol \( \circ \) signifies function composition. \( (f \circ g)(x) \) means applying \( g(x) \) first, then using that result as input for \( f(x) \).
• **Input Substitution**: While computing \((f \circ g)(x)\), substitute \( g(x) \) into \( f(x) \), leading to \( f(g(x)) = (x - 1)^2 \).
• **Reversed Composition**: Conversely, \((g \circ f)(x)\) indicates using \( f(x) \) as the input to \( g(x) \), which becomes \( g(f(x)) = x^2 - 1 \).

Understanding this notation allows you to efficiently decode the process of solving complex function interactions and compositions.