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Chapter 9: Problem 9

$$ f(x)=3^{5 x} $$

Short Answer

Expert verified
Evaluate the function \( f(x) = 3^{5x} \) by substituting the value of \( x \) in the exponent.

Step by step solution

01

- Understand the Function

The given function is an exponential function of the form \( f(x) = 3^{5x} \).
02

- Identify the Base and the Exponent

In the function \( f(x) = 3^{5x} \), the base is 3 and the exponent is \( 5x \).
03

- Evaluate the Function at Given Points

To understand the behavior of the function, evaluate it at certain points. For example, when \( x = 0 \): \( f(0) = 3^{5 \times 0} = 3^0 = 1 \).
04

- General Evaluation

To evaluate the function at any general point \( x \), simply plug in \( x \) into the exponent part of the function. For example:\( f(2) = 3^{5 \times 2} = 3^{10} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Base
The base of an exponential function is a crucial element that defines the overall behavior of the function. In the exponential function given, \( f(x) = 3^{5x} \), the base is 3. The base is the number that is raised to the power of the exponent. Here are some important points about the base:
  • The base must be a positive real number if we are dealing with real-valued functions.
  • A base greater than 1, like 3 in our given function, indicates exponential growth. The function values increase rapidly as the exponent increases.
  • If the base were between 0 and 1, it would indicate exponential decay, where the function values decrease rapidly.
  • When the base is exactly 1, the function becomes constant because any power of 1 is always 1.
Understanding the base helps us predict how the function behaves as the exponent changes.
Exponent
In an exponential function, the exponent determines the rate at which the function grows or decays. For the given function \( f(x) = 3^{5x} \), the exponent is \( 5x \). This part of the function deserves special attention for several reasons:
  • Unlike the base, which is constant, the exponent can be a variable expression, here it's \( 5x \).
  • The higher the exponent, the faster the function value will rise or fall. Since the exponent in our case is 5 times the variable \( x \), small changes in \( x \) can result in very large changes in \( f(x) \).
  • If the exponent is negative, it will make the base a reciprocal. For example, if \( x = -1 \), then \( 5x = -5 \) and the function becomes \( f(-1) = 3^{-5} = \frac{1}{3^5} \).
  • If the exponent is zero, then any non-zero base raised to the power of zero equals 1.
Understanding the exponent helps you manipulate and evaluate the function at different points.
Function Evaluation
Evaluating an exponential function means finding its value when the input variable \( x \) is given specific values. For our function \( f(x) = 3^{5x} \), here's how you can evaluate it step-by-step:
  • Step 1: Identify the input value for \( x \). For example, let鈥檚 take \( x = 2 \).
  • Step 2: Substitute \( x \) into the exponent. This gives us \( f(2) = 3^{5 \times 2} = 3^{10} \).
  • Step 3: Compute the result. Here, \( 3^{10} \) is a large number, specifically 59,049. Notice how quickly the function values ramp up due to the exponential nature.
Bringing it all together, whether you're given \( x = 0 \), where \( f(0) = 1 \) because any number raised to the power of zero is 1, or any other value, just apply these steps to evaluate the function accurately. Practicing this will help reinforce your understanding of exponential functions and their evaluations.

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Most popular questions from this chapter

Find the area of the region bounded by the curve \(y=2 /(x-3)\), the \(x\) axis, and the lines \(x=4\) and \(x=5\).

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