Problem 19 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

(a) -11, (b) -119

Step 1: Understand Function Composition

Function composition involves applying one function to the results of another function. Given two functions \( f(x) \) and \( g(x) \), you can compose them as \((f \circ g)(x) = f(g(x))\) and \((g \circ f)(x) = g(f(x))\).

Step 2: Evaluate \( g(-2) \)

To evaluate \((f \circ g)(-2)\), first find \(g(-2)\). Use the function \(g(x) = 2 - x^2\). Substitute \(x = -2\) into the function: \[g(-2) = 2 - (-2)^2 = 2 - 4 = -2\].

Step 3: Evaluate \( f(g(-2)) \)

Now that we have \(g(-2) = -2\), use this value to evaluate \(f(g(-2)) = f(-2)\). Use the function \(f(x) = 3x - 5\). Substitute \(x = -2\): \[f(-2) = 3(-2) - 5 = -6 - 5 = -11\].

Step 4: Evaluate \( f(-2) \)\ for \((g \circ f)(-2)\)

Now evaluate \(f(-2)\) again for \((g \circ f)(-2)\), where \(f(x) = 3x - 5\). We already know \(f(-2) = -11\).

Step 5: Evaluate \( g(f(-2)) \)

Use the result from Step 4. Evaluate \(g(f(-2)) = g(-11)\) using \(g(x) = 2 - x^2\). Substitute \(x = -11\): \[g(-11) = 2 - (-11)^2 = 2 - 121 = -119\].

Key Concepts

Composite Functions

In mathematics, when you combine two functions to form a new function, it is called a composite function. Imagine it like a machine that processes inputs through two stages. First, the input goes through one function, and the result becomes the input for the second function. This is written as

• \((f \circ g)(x) = f(g(x))\) which means that the output of function \(g\) is used as the input for function \(f\).
• Similarly, \((g \circ f)(x) = g(f(x))\).

This way, you get a single output from two connected functions. It's like putting something through two stages of processing.
Understanding this process helps not just in math, but in logical thinking about sequences and order of operations. In the given example, we have functions \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\). We can find

• \((f \circ g)(-2)\)
• and \((g \circ f)(-2)\)

by applying the operations step by step. Remember, the order of functions matters greatly in composition, as switching them around gives different results!

Function Evaluation

Function evaluation means finding the value of a function at a particular input. This is like feeding the machine an input and watching the output it produces from its calculations. To start, identify the function you're dealing with and the specific input value given.
For example, if we are working with \(f(x) = 3x - 5\), then to evaluate \(f(-2)\), substitute \(-2\) for \(x\) in the expression. This gives us:

• \[f(-2) = 3(-2) - 5 = -6 - 5 = -11\].

Evaluating \(g(x) = 2 - x^2\) at \(-2\) involves, similarly, substituting into the function to get:

• \[g(-2) = 2 - (-2)^2 = 2 - 4 = -2\].

Function evaluation is essential for understanding how functions behave for different inputs and plays a crucial role in analyses and predictions in math and sciences.

Algebraic Functions

Algebraic functions are those built from polynomial, rational, root, or even trigonometric pieces using algebraic operations—such as addition, subtraction, multiplication, division, and taking roots. They are everywhere in math, providing ways to describe curves, models, and systems with precision.
For instance, in the functions we’re dealing with here:

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

Both are algebraic because they’re made from variables and constants with operations like addition, subtraction, and exponentiation.
These functions are used to solve equations and model real-world phenomena, such as calculating trajectories, predicting market trends, or evaluating scientific data. Having a firm grasp of algebraic functions is vital as it equips you with tools to tackle a wide array of problems both academic and practical.