Problem 8 - Precalculus Mathematics for Calculus

Question

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$g(x)=\left(\frac{3}{4}\right)^{2 x} ; \quad g(0.7), g(\sqrt{7} / 2), g(1 / \pi), g\left(\frac{2}{3}\right)$$

Step-by-Step Solution

Answer

g(0.7) ≈ 0.686, g(\(\sqrt{7}/2\)) ≈ 0.438, g(1/\(\pi\)) ≈ 0.858, g(2/3) ≈ 0.707.

Step 1: Evaluate for g(0.7)

To evaluate \(g(0.7)\), substitute \(x = 0.7\) into the function: \[g(0.7) = \left(\frac{3}{4}\right)^{2 \times 0.7} = \left(\frac{3}{4}\right)^{1.4}\]. Using a calculator, compute \(\left(\frac{3}{4}\right)^{1.4} \approx 0.686\).

Step 2: Evaluate for g(\(\sqrt{7} / 2\))

Substitute \(x = \sqrt{7}/2\) into the function: \[g\left(\frac{\sqrt{7}}{2}\right) = \left(\frac{3}{4}\right)^{2 \times \frac{\sqrt{7}}{2}} = \left(\frac{3}{4}\right)^{\sqrt{7}}\]. Calculate \(\left(\frac{3}{4}\right)^{\sqrt{7}} \approx 0.438\) using a calculator.

Step 3: Evaluate for g(1/\(\pi\))

Substitute \(x = 1/\pi\) into the function: \[g\left(\frac{1}{\pi}\right) = \left(\frac{3}{4}\right)^{2 \times \frac{1}{\pi}} = \left(\frac{3}{4}\right)^{2/\pi}\]. Use a calculator to find \(\left(\frac{3}{4}\right)^{2/\pi} \approx 0.858\).

Step 4: Evaluate for g(2/3)

Substitute \(x = 2/3\) into the function: \[g\left(\frac{2}{3}\right) = \left(\frac{3}{4}\right)^{2 \times \frac{2}{3}} = \left(\frac{3}{4}\right)^{4/3}\]. Evaluate \(\left(\frac{3}{4}\right)^{4/3} \approx 0.707\) using a calculator.

Key Concepts

Evaluating functions

Evaluating functions is all about finding the output of a function given an input. Functions are like machines; you put something in, it processes it according to a rule, and out comes the result. In mathematical terms, a function is often represented by a formula with a variable, such as \( g(x) = \left(\frac{3}{4}\right)^{2x} \). To evaluate \( g(x) \) at particular values (like \(0.7, \frac{\sqrt{7}}{2}, \frac{1}{\pi}, \) and \(\frac{2}{3}\)), follow these steps:- **Substitute the variable:** Replace \( x \) with the specific number you want to evaluate in the function.- **Simplify the expression:** Do any necessary arithmetic to simplify the power/exponent calculations.Every function has a unique rule, and evaluating it correctly involves sticking to that rule and the order of operations. Remember, the goal is to find out what the function spits out for any given input. This process is both systematic and follows repetitive steps each time.

Calculator usage in mathematics

Using a calculator effectively in mathematics can make your life much easier. Especially in problems involving complex computations and irrational numbers, calculators are indispensable. They can quickly calculate exponents, roots, and other operations that would take much longer by hand.

• **Fractional exponents:** Enter the base first, then use the power or exponent key to input fractional exponents. Make sure to use parentheses to ensure correct inputs.
• **Rounding:** After obtaining your result, round it to the necessary decimal places. For calculations in this exercise, we round to three decimal places.
• **Complex inputs:** For expressions like square roots or division fractions, ensure correct inputs using parentheses to clarify the order of operations.

Calculators are powerful when used correctly, and practice can significantly improve your accuracy and efficiency. Ensure you're familiar with your calculator's functions, and take the time to verify results when possible.

Rational exponents

Rational exponents combine the ideas of roots and powers and provide an efficient way to express complex operations. A rational exponent is an exponent that is a fraction rather than an integer. For example, the expression \( \left(\frac{3}{4}\right)^{1.4} \) represents a number raised to a rational exponent, where 1.4 is the exponent in decimal form.
Rational exponents follow the same rules as integer exponents:

• **Product of powers:** Multiply like bases by adding their exponents.
• **Quotient of powers:** Divide like bases by subtracting the exponents.
• **Power of a power:** Raise a power to another power by multiplying the exponents.

When dealing with rational exponents, you can often rewrite them using roots. For instance, \( n^{1/2} \) is the same as the square root of \( n \). Understanding how to switch between these forms can simplify calculations, especially when using a calculator, and is useful when evaluating functions with complex expressions.