Problem 26 - Algebra and Trigonometry

Question

Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{ll}{\text { (a) }(f \circ f)(x)} & {\text { (b) }(g \circ g)(x)}\end{array} $$

Step-by-Step Solution

Answer

\((f \circ f)(x) = 9x - 20\); \((g \circ g)(x) = -2 + 4x^2 - x^4\).

Step 1: Understanding Function Composition

In this task, we need to evaluate the expressions \((f \circ f)(x)\) and \((g \circ g)(x)\). The notation \((f \circ f)(x)\) means substituting \(f(x)\) into itself, and \((g \circ g)(x)\) means substituting \(g(x)\) into itself.

Step 2: Calculate \((f \circ f)(x)\)

First, compute \(f(f(x))\) by substituting \(f(x)\) into itself. We know \(f(x) = 3x - 5\), so substituting it gives:\[ f(f(x)) = f(3x - 5) \]Now apply the function rule \(f(x) = 3x - 5\) to \(3x - 5\): \[ f(3x - 5) = 3(3x - 5) - 5 \]Simplify by distributing the \(3\):\[ = 9x - 15 - 5 = 9x - 20 \]Thus, \((f \circ f)(x) = 9x - 20\).

Step 3: Calculate \((g \circ g)(x)\)

Next, calculate \(g(g(x))\) by substituting \(g(x)\) into itself. We know \(g(x) = 2 - x^2\), so substituting it gives:\[ g(g(x)) = g(2 - x^2) \]Apply the function rule \(g(x) = 2 - x^2\) to \(2 - x^2\):\[ g(2 - x^2) = 2 - (2 - x^2)^2 \]Now, expand \((2 - x^2)^2\):\[ (2 - x^2)^2 = (2 - x^2)(2 - x^2) = 4 - 4x^2 + x^4 \] Substitute back:\[ = 2 - 4 + 4x^2 - x^4 = -2 + 4x^2 - x^4 \]So, \((g \circ g)(x) = -2 + 4x^2 - x^4\).

Key Concepts

Algebraic Functions

Algebraic functions are expressions built using algebraic operations like addition, subtraction, multiplication, and division, along with exponentiation involving real numbers. These functions describe the relationship between variables in a mathematical context. In this exercise, we are dealing with two algebraic functions, namely:

• \( f(x) = 3x - 5 \)
• \( g(x) = 2 - x^2 \)

These functions involve simple mathematical operations that highlight the nature of algebraic functions. The function \( f(x) \) is linear, characterized by its highest exponent (1), while \( g(x) \) is a quadratic function, marked by an exponent of 2. Each function outputs a value based on the input \( x \), using its specific algebraic expression. Understanding these functions' forms helps comprehend more complex operations like function composition.

Substitution Method

The substitution method is a key tool in solving equations and evaluating expressions, particularly when dealing with function composition. This method involves replacing one expression with another equivalent expression.In the context of function composition, substitution occurs when one function is placed into another. For instance, in our exercise:

• \((f \circ f)(x)\) translates to substituting \(f(x)\) back into itself.
• \((g \circ g)(x)\) means substituting \(g(x)\) into itself.

To do this, you replace the variable \(x\) in the outer function with the entire expression of the inner function. For example, evaluating \(f(f(x))\) means replacing \(x\) in \(f(x) = 3x - 5\) with \(f(x)\) itself, i.e., \(3x - 5\). This results in \(f(3x - 5)\), which is further simplified using algebraic rules.Using the substitution method simplifies complex expressions and is crucial for understanding how different functions interact when composed together.

Function Evaluation

Function evaluation involves calculating the output of a function for a given input. It is a fundamental concept in mathematics that applies to various operations involving functions.In our case, you evaluated both \((f \circ f)(x)\) and \((g \circ g)(x)\) by applying each function to the outcome of the substitution. This process essentially involves multiple steps:

• Substitute the given expression into the function, converting the statement into a solvable equation.
• Simplify the expression using appropriate arithmetic operations like distribution and subtraction.
• A new, simplified expression yields the function's evaluation.

For \((f \circ f)(x)\), after substituting, you simplify \(f(3x - 5)\) further to find the composition result as \(9x - 20\). Similarly, resolving \((g \circ g)(x)\) involves substituting and simplifying \(g(2 - x^2)\), ultimately resulting in \(-2 + 4x^2 - x^4\).Looking at the functional transformation of inputs and their respective outputs helps develop intuition about functions and their behaviors, a crucial skill in algebra.