Problem 21 - Algebra and Trigonometry

Question

Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{ll}{\text { (a) } f(g(0))} & {\text { (b) } g(f(0))}\end{array} $$

Step-by-Step Solution

Answer

(a) 1, (b) -23

Step 1: Find g(0)

First, substitute 0 into the function \( g(x) = 2 - x^2 \). This will give us \( g(0) = 2 - 0^2 = 2 \).

Step 2: Find f(g(0))

Now that we know \( g(0) = 2 \), substitute 2 into the function \( f(x) = 3x - 5 \). This gives us \( f(2) = 3 \times 2 - 5 = 6 - 5 = 1 \).

Step 3: Find f(0)

Next, substitute 0 into the function \( f(x) = 3x - 5 \). This yields \( f(0) = 3 \times 0 - 5 = -5 \).

Step 4: Find g(f(0))

Now, substitute \( -5 \) into the function \( g(x) = 2 - x^2 \). This gives us \( g(-5) = 2 - (-5)^2 = 2 - 25 = -23 \).

Key Concepts

Evaluating Functions

When we talk about evaluating functions, we mean finding the output for a specific input value. Imagine a machine where you insert a number (input) and get another number (output) after some processing. Functions do exactly that. For the given function, such as \(f(x) = 3x - 5\), evaluating the function means replacing the \(x\) with a specific value to find the result.
This process involves substituting every occurrence of \(x\) with your chosen number and performing the arithmetic operations as the machine (or function formula) describes.

Consider the function composition we have here: To evaluate \(f(g(0))\), you first find what \(g(0)\) is. Then, use this result and plug it into \(f(x)\). It's like a layered processing machine, first using \(g(x)\) and then \(f(x)\) based on \(g\)'s output.

• Step 1: Evaluate the inner function
• Step 2: Use the result to evaluate the outer function

Algebraic Functions

Algebraic functions are expressions that involve polynomials. They can include operations such as addition, subtraction, multiplication, division, and even taking roots. In this exercise, \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) are both algebraic functions.
Both functions involve polynomial terms, with the highest power of \(x\) being 1 in \(f(x)\) (making it linear) and 2 in \(g(x)\) (making it quadratic).

It's important to understand the nature of these functions because it helps in predicting the shape of their graphs and comprehending their behavior through various inputs.

• Linear function: Typically forms a straight line.
• Quadratic function: Forms a parabolic curve

Knowing this assists in understanding more complex algebraic structures and can aid in seeing how changes in input affect outputs.

Function Notation

Function notation is a way of writing used to denote functions clearly. It helps to identify which input is being considered and integrates seamlessly into mathematical expressions.
In the notations \(f(x)\) and \(g(x)\), \(f\) and \(g\) are the function names and \(x\) is the input variable. This notation helps streamline communication in mathematics, indicating the operation being performed on \(x\).

Using function notation, one can communicate complex ideas clearly:

• \(f(g(0))\) shows composition, using \(G(x)\) then \(F(x)\).
• \(g(f(0))\) does the reverse, starting with \(F(x)\).

Finally, function notation allows you to easily express and solve for outputs across different contexts, and provides a method to understand algebraic manipulations in a clear, consistent way.