Problem 95 - College Algebra Essentials

Question

let \(f\) and \(g\) be defined by the following table: $$ \begin{array}{ccc} \hline x & f(x) & g(x) \\ \hline-2 & 6 & 0 \\ -1 & 3 & 4 \\ 0 & -1 & 1 \\ 1 & -4 & -3 \\ 2 & 0 & -6 \end{array} $$ Find \(\sqrt{f(-1)-f(0)}-[g(2)]^{2}+f(-2) \div g(2) \cdot g(-1)\)

Step-by-Step Solution

Answer

The result of the calculation is -38.

Step 1: Find Function Values

First, let's look up in the tables to find the right values. \(f(-1)\) is 3, \(f(0)\) is -1, \(g(2)\) is -6, \(f(-2)\) is 6 and \(g(-1)\) is 4.

Step 2: Substitute and compute

Now that we have all the values, we can substitute them in the equation: \(\sqrt{f(-1)-f(0)}-[g(2)]^{2}+f(-2) \div g(2) \cdot g(-1)\equiv \sqrt{3-(-1)}-(-6)^2+6\div -6 \cdot 4\) which simplifies to \(\sqrt{4}-36+(-1 \cdot 4)\). Now apply the square root, which gives us \(2-36-4\).

Step 3: Final Calculation

The final step is to make the calculation from the previous step: \(2-36-4 \equiv -38\)

Key Concepts

Function Table

When working with functions, a function table is an excellent tool for organizing data and seeing how each function behaves for different input values. In our exercise, we have two functions: \( f(x) \) and \( g(x) \). The table provides specific values of these functions at certain \( x \)-points.
For example, if you want to know the value of \( f(-1) \), you can simply look it up in the table instead of calculating it. This allows us to quickly identify function values without the need for complex calculations. Here, from the table:

• \( f(-1) = 3 \)
• \( f(0) = -1 \)
• \( g(2) = -6 \)
• \( f(-2) = 6 \)
• \( g(-1) = 4 \)

Using these values, we can perform various operations to solve the problem.

Square Root

The square root is a basic mathematical operation that finds a number which, when multiplied by itself, produces the original number. In other words, if \( \sqrt{4} = 2 \), it means that the number 2 multiplied by itself equals 4.
In our problem, we need to find \( \sqrt{f(-1)-f(0)} \). This means we substitute the values of \( f(-1) \) and \( f(0) \) to get \( \sqrt{3 - (-1)} \). Since \( 3 - (-1) \) equals 4, the square root is simply 2.
Applying the square root operation simplifies part of our equation and helps us move towards finding the final answer.

Substitute and Compute

Substituting means replacing the function identifiers with their actual values. This is essential for solving the equation. From our table, we've identified the necessary function values. Our task is to insert these into our expression and then compute the result.
Here's how it's done:

• Replace \( f(-1) \) with 3, \( f(0) \) with -1, \( g(2) \) with -6, \( f(-2) \) with 6, and \( g(-1) \) with 4.

Now your expression will be: \( \sqrt{3 - (-1)} - [(-6)]^2 + 6 \div (-6) \cdot 4 \).
Carry out these calculations step-by-step. Start with computing the square of \(-6\) which gives 36, and then calculate \(6 \div (-6) \cdot 4 \) which equals -4. Substituting these results into our formula brings us closer to the final solution.

Arithmetic Operations on Functions

Performing arithmetic operations on function outputs involves basic operations such as adding, subtracting, multiplying, and dividing function values. This component of the problem features these operations and shows how functions can be combined and manipulated.
In our given problem, we have:

• A square root operation: \( \sqrt{4} \) which results in 2.
• Subtracting the square of \( g(2) \) (i.e., \(-36\)) from the result.
• Adding the product of \( f(-2) \div g(2) \) and \( g(-1) \) (i.e., \(-4\)).

Combine these results with the respective operations:- Start with 2 (from the square root)- Then subtract 36- Finally, subtract another 4
This gives you the final result of the function operation as -38. Understanding this component is essential as arithmetic operations on functions form a foundation for more complex mathematical concepts.