Problem 17 - Algebra and Trigonometry

Question

\(17-22=\) Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{ll}{\text { (a) } f(g(0))} & {\text { (b) } g(f(0))}\end{array} $$

Step-by-Step Solution

Answer

(a) 1, (b) -23.

Step 1: Substitute in Function g

Start with finding \(g(0)\). You substitute \(x = 0\) into \(g(x) = 2 - x^2\). This gives \(g(0) = 2 - 0^2 = 2\).

Step 2: Use Result in Function f

Now we need to find \(f(g(0))\), which is \(f(2)\) because \(g(0) = 2\). Substitute \(x = 2\) into \(f(x) = 3x - 5\). This gives \(f(2) = 3(2) - 5 = 6 - 5 = 1\).

Step 3: Substitute in Function f

Next, find \(f(0)\). Substitute \(x = 0\) into \(f(x) = 3x - 5\). This results in \(f(0) = 3(0) - 5 = -5\).

Step 4: Use Result in Function g

Now we need \(g(f(0))\), which is \(g(-5)\) because \(f(0) = -5\). Substitute \(x = -5\) into \(g(x) = 2 - x^2\). This gives \(g(-5) = 2 - (-5)^2 = 2 - 25 = -23\).

Key Concepts

Algebra

Algebra is like a toolkit for solving mathematical problems using symbols and letters to represent numbers and quantities. This branch of mathematics is essential for understanding the relationship between variables and solving equations. In our exercise, algebra helps us understand how to work with the functions \(f(x)\) and \(g(x)\) by substituting values and using operations like addition, subtraction, and multiplication. Instead of calculating large expressions manually, algebra allows us to create a general formula and use it to streamline solutions.

Functions

Functions are like machines where you input a number, perform operations, and get another number as output. Each function has its specific rule. In our exercise, we have two distinct functions:

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

The variable \(x\) inside each function is like an empty slot that can be filled by any number. Think of \(f\) and \(g\) as two machines each following their unique process. In \(f(x)\), every \(x\) gets tripled and then reduced by five. For \(g(x)\), \(x\) is squared, and that square is subtracted from two.
By using these operations, functions let you explore how changes to \(x\) affect the outcome. Functions are fundamental in understanding and visualizing mathematical relationships. Understanding these properties makes it easier to predict and calculate outputs when functions are applied to different inputs.

Substitution Method

The substitution method is a technique for replacing a variable with its corresponding value or expression within a function. This tool is crucial for solving problems involving complex equations where multiple variables interact. In our task, we used substitution to find \(f(g(0))\) and \(g(f(0))\).
First, we substituted \(x = 0\) into \(g(x)\) to get \(g(0)\), simplifying it step by step. Then, we used the result to find \(f(g(0))\), by placing \(g(0)\)'s value into function \(f\).
Later, we operated similarly for \(g(f(0))\), starting with initial substitution in \(f\) and proceeding to \(g\). This method is about direct replacement, which simplifies equations through systematic reduction. Understanding the substitution method helps in breaking down multi-step problems into simpler, more manageable pieces, allowing for efficient problem-solving in algebra and beyond.