Problem 20 - Algebra and Trigonometry

Question

\(17-22=\) Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{ll}{\text { (a) }(f \circ f)(-1)} & {\text { (b) }(g \circ g)(2)}\end{array} $$

Step-by-Step Solution

Answer

(a) -29 (b) -2

Step 1: Simplify the Expression 17-22

First, we perform the subtraction to simplify the expression:\[17 - 22 = -5\]

Step 2: Define the Functions f(x) and g(x)

We have two functions given:- \( f(x) = 3x - 5 \)- \( g(x) = 2 - x^2 \)

Step 3: Evaluate the Composition (f ∘ f)(-1)

To find \((f \circ f)(-1)\), we need to evaluate \(f(f(-1))\).- First, compute \(f(-1)\):\[ f(-1) = 3(-1) - 5 = -3 - 5 = -8 \]- Next, compute \(f(-8)\):\[ f(-8) = 3(-8) - 5 = -24 - 5 = -29 \]Thus, \((f \circ f)(-1) = -29\).

Step 4: Evaluate the Composition (g ∘ g)(2)

To find \((g \circ g)(2)\), we need to evaluate \(g(g(2))\).- First, compute \(g(2)\):\[ g(2) = 2 - 2^2 = 2 - 4 = -2 \]- Next, compute \(g(-2)\):\[ g(-2) = 2 - (-2)^2 = 2 - 4 = -2 \]Thus, \((g \circ g)(2) = -2\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. It’s like inputting a number into a machine and getting a specific result. For instance, for the function \(f(x) = 3x - 5\), you can evaluate the function at \(x = -1\) by substituting \(-1\) into the function.

• Plug in the value: \(f(-1) = 3(-1) - 5\).
• Calculate the result: \(-3 - 5\), which gives \(-8\).

Once you substitute and simplify, you've successfully evaluated the function at that given point. You can use this fundamental skill for more complex operations like composite functions, where you'll also evaluate one function, get its result, and plug it into another function.

Algebraic Functions

Algebraic functions are expressions that involve mathematical operations such as addition, subtraction, multiplication, division, and exponentiation. An algebraic function can be something as simple as \(f(x) = 3x - 5\) or as intricate as polynomial functions.

• Linear functions have the form \(ax + b\).
• Quadratic functions are expressed as \(ax^2 + bx + c\).

Our examples, \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\), show different types of algebraic functions. Knowing how to use these functions involves recognizing their components and effectively performing operations to evaluate them. Mastery of this will help you understand more advanced topics, including composite functions.

Composite Functions

Composite functions involve creating a new function by applying one function to the result of another. This concept can be thought of as "function within a function." Given two functions \(f(x)\) and \(g(x)\), the composition \((f \circ g)(x)\) means you substitute \(g(x)\) into \(f(x)\).

• First, evaluate \(g(x)\) to get a result.
• Then substitute this result into \(f(x)\) if you’re calculating \((f \circ g)(x)\), or vice versa.

In the problem, you're asked to find \((f \circ f)(-1)\) and \((g \circ g)(2)\). This means you need to evaluate \(f(x)\) or \(g(x)\) twice, each time using the output of the previous evaluation. It's a crucial skill when working through complex function compositions like these.