Problem 2 - Contemporary Precalculus

Question

Find the indicated values of the function by hand and by using the table feature of a calculator (or the EVAL key on TI-85/86). If your answers do not agree with each other or with those at the back of the book, you are either making algebraic mistakes or incorrectly entering the function in the equation memory. \(g(x)=\sqrt{x+4}-2\) (b) \(g(0)\) (c) \(g(4)\) (a) \(g(-2)\) (d) \(g(5)\) (e) \(g(12)\)

Step-by-Step Solution

Answer

Question: Calculate the value of the function \(g(x)=\sqrt{x+4}-2\) for \(x = 0\), \(x = -2\), \(x = 4\), \(x = 5\), and \(x = 12\) both by hand and using a calculator. Verify that the results are consistent. Answer: - When \(x = 0\), \(g(0) = \sqrt{0+4}-2 = 0\). - When \(x = -2\), \(g(-2) = \sqrt{-2+4}-2 = \sqrt{2}-2\). - When \(x = 4\), \(g(4) = \sqrt{4+4}-2 = \sqrt{8}-2\). - When \(x = 5\), \(g(5) = \sqrt{5+4}-2 = 1\). - When \(x = 12\), \(g(12) = \sqrt{12+4}-2 = 2\). By substituting the above values of x in the function on the calculator, we get the same result, hence confirming that the hand calculations are correct.

Step 1: Find g(0) by hand

To find the value of the function at x=0, plug in x=0 into the equation: \(g(0)=\sqrt{0+4}-2\) \(g(0)=\sqrt{4}-2\) \(g(0)=2-2\) \(g(0)=0\)

Step 2: Find g(0) using a calculator

Input the function into your calculator and set x=0. Verify that the result agrees with the hand-calculated value (\(g(0)=0\)).

Step 3: Find g(-2) by hand

To find the value of the function at x=-2, plug in x=-2 into the equation: \(g(-2)=\sqrt{-2+4}-2\) \(g(-2)=\sqrt{2}-2\)

Step 4: Find g(-2) using a calculator

Input the function into your calculator and set x=-2. Verify that the result agrees with the hand-calculated value (\(g(-2)=\sqrt{2}-2\)).

Step 5: Find g(4) by hand

To find the value of the function at x=4, plug in x=4 into the equation: \(g(4)=\sqrt{4+4}-2\) \(g(4)=\sqrt{8}-2\)

Step 6: Find g(4) using a calculator

Input the function into your calculator and set x=4. Verify that the result agrees with the hand-calculated value (\(g(4)=\sqrt{8}-2\)).

Step 7: Find g(5) by hand

To find the value of the function at x=5, plug in x=5 into the equation: \(g(5)=\sqrt{5+4}-2\) \(g(5)=\sqrt{9}-2\) \(g(5)=3-2\) \(g(5)=1\)

Step 8: Find g(5) using a calculator

Input the function into your calculator and set x=5. Verify that the result agrees with the hand-calculated value (\(g(5)=1\)).

Step 9: Find g(12) by hand

To find the value of the function at x=12, plug in x=12 into the equation: \(g(12)=\sqrt{12+4}-2\) \(g(12)=\sqrt{16}-2\) \(g(12)=4-2\) \(g(12)=2\)

Step 10: Find g(12) using a calculator

Input the function into your calculator and set x=12. Verify that the result agrees with the hand-calculated value (\(g(12)=2\)).

Key Concepts

Calculators in Mathematics

Calculators are powerful tools that assist us in quickly finding values and verifying solutions. When working with functions, calculators can perform calculations accurately and efficiently, saving time and reducing human errors.
For tasks like evaluating functions, the table feature or functions such as the EVAL key on calculators like the TI-85/86 can be especially useful. To use this feature, you enter the expression of the function and specify the value of the independent variable, in this case, the value of \(x\).
Calculators can help check work done manually. This ensures that results align with expected values and catch any arithmetic mistakes.

• Without calculators, you primarily rely on hand calculations, which can often be cumbersome for complex functions.
• It's important to correctly input the entire function equation to get accurate outputs. A misplaced parenthesis or a wrong operation sign can lead to incorrect results.

Consequently, calculators not only aid in computation but also serve as a means to verify our algebraic work.

Solving Functions

When solving functions, the primary goal is to evaluate the function at specific points. Take, for example, the function given \(g(x)=\sqrt{x+4}-2\). We need to find \(g(0)\), \(g(4)\), and so on by substituting values into the function.
The process involves:

• Rewriting the function at a particular value of \(x\).
• Perform the arithmetic operations step by step.
• Square root any possible values before subtracting, thus following the order of operations to avoid errors.

Let's demonstrate with \(g(0)\):
1. Substitute \(x=0\) into the function gives us \(g(0)=\sqrt{0+4}-2\).
2. Simplify inside the square root, so \(\sqrt{4}-2\).
3. Calculate the square root: \(2-2\), resulting in \(0\).
As we can see, solving functions helps ensure clarity and accuracy, especially when done step by step.

Algebraic Verification

Algebraic verification is the process of confirming that the manual computations align with calculator results. This is crucial for ensuring that calculations are error-free and logically sound.
For example, after calculating \(g(0)\) manually, you would use a calculator to enter the function and check that it outputs the same result, thus verifying correctness.
This step is invaluable because:

• It builds confidence in the solution, knowing that both methods lead to the same answer.
• It helps identify errors that might have occurred during manual calculations.

Verifying each calculated function value is a practical approach to mastering calculations. It ensures that your understanding of functions and their manipulations is correct, thus reinforcing learning in both computational and algebraic methods.