Problem 2 - Algebra and Trigonometry

Question

1–4 ? Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$ f(x)=3^{x+1} ; \quad f(-1.5), f(\sqrt{3}), f(e), f\left(-\frac{5}{4}\right) $$

Step-by-Step Solution

Answer

The function values are approximately: 0.577, 13.841, 73.944, and 0.759.

Step 1: Understanding the Function

We need to evaluate the function \( f(x) = 3^{x+1} \) for different values of \( x \). The formula means that we will raise 3 to the power of \( x+1 \) for each given \( x \).

Step 2: Evaluating f(-1.5)

Replace \( x \) with \( -1.5 \) in the function: \( f(-1.5) = 3^{-1.5+1} = 3^{-0.5} \). Use a calculator to find \( 3^{-0.5} \), and round to three decimals to get approximately 0.577.

Step 3: Evaluating f(\sqrt{3})

Replace \( x \) with \( \sqrt{3} \) in the function: \( f(\sqrt{3}) = 3^{\sqrt{3}+1} \). First, calculate \( \sqrt{3} \), which is approximately 1.732. Thus, \( f(\sqrt{3}) = 3^{2.732} \). Use a calculator to find \( 3^{2.732} \) and round to three decimals, resulting in approximately 13.841.

Step 4: Evaluating f(e)

Replace \( x \) with \( e \) (Euler's number) in the function: \( f(e) = 3^{e+1} \). First, note \( e \approx 2.718 \). Then, calculate \( 3^{3.718} \) using a calculator. The result, rounded to three decimals, is approximately 73.944.

Step 5: Evaluating f\left(-\frac{5}{4}\right)

Replace \( x \) with \( -\frac{5}{4} \) in the function: \( f\left(-\frac{5}{4}\right) = 3^{-\frac{5}{4}+1} = 3^{-\frac{1}{4}} \). Use a calculator to find \( 3^{-0.25} \) and round to three decimals to get approximately 0.759.

Key Concepts

Function Evaluation

Evaluating a function is an important skill in mathematics. It involves finding the value of the function for specific input values. Consider the function described here: \( f(x) = 3^{x+1} \). This function tells us that for any given \( x \), we need to do a series of steps to find \( f(x) \):

• Add 1 to the given \( x \). This changes the exponent that 3 needs to be raised to.
• Raise 3 to the resulting power. This means performing exponentiation, a mathematical operation where a number is multiplied by itself a specific number of times.
• Once you find the result, you have successfully evaluated the function at that point.

This skill is incredibly useful as functions can be used to describe many real-world scenarios, from calculating interest to predicting growth patterns.

Rounding Decimals

Rounding decimals is an essential mathematical practice, especially when dealing with functions and large calculations. Often, calculations result in long decimal numbers, which can be cumbersome to handle. So, let's see why and how we round decimals:

• Why Round? Rounding makes numbers easier to work with, requiring less storage in computer systems and less space in our brains.
• How to Round: Decide how many decimal places you want (here, we're rounding to three decimal places). Look at the number in the next decimal place. If it's 5 or more, increase the last digit you're keeping by one.

For example, if you have the number 0.57735, after rounding it to three decimal places, you get 0.577. This helps convey a sound and practical approximation of the value, which is often more relevant and meaningful in solutions.

Calculator Usage

Calculators are vital tools when performing exponentiation and complex mathematical computations. Knowing how to use them effectively ensures precision and saves time. Here’s how to make the most of your calculator when evaluating functions like \( f(x) = 3^{x+1} \):

• Understand your calculator's power function. Most calculators have a \( y^x \) button or similar to assist with exponentiation.
• Input the base number first (in this case, 3), then use the power function key, followed by the exponent value.
• When dealing with mathematical constants like \( e \), many calculators have a dedicated \( e \) button, allowing for ease in handling exponential functions.

Accurate calculator usage leads to precise function evaluation, ensuring your results are correct given your inputs.

Exponentiation

Exponentiation is one of the core operations in mathematics, akin to addition and multiplication. It's the process of raising a number (the base) to the power of another number (the exponent). Understanding this can be crucial, especially in tackling exercises like \( f(x) = 3^{x+1} \).

• Exponents indicate how many times the base is multiplied by itself. For instance, \( 3^2 \) means \( 3 \times 3 \).
• When dealing with fractional or decimal exponents, we get roots. For instance, \( 3^{0.5} \) gives the square root of 3.
• Negative exponents result in the reciprocal of the base raised to the absolute value of the exponent, like \( 3^{-1} = \frac{1}{3} \).

Mastering exponentiation expands your ability to solve various mathematical problems, from simple calculations to more complex modeling scenarios.