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Chapter 4: Problem 4

Fill in the blank. The domain of \(f(x)=a^{x}\) for \(a>0\) is the set of _________.

Short Answer

Expert verified
The domain of \(f(x)=a^x\) for \(a>0\) is the set of all real numbers.

Step by step solution

01

Understand the function

The function given is an exponential function: \(f(x) = a^x\) where \(a\) is a positive constant.
02

Identify what the domain represents

The domain of a function represents all the possible input values (x-values) for which the function is defined.
03

Determine the domain for the exponential function

An exponential function \(f(x) = a^x\) is defined for all real numbers, meaning any real number can be plugged in for \(x\) and the function will produce a valid output.
04

State the domain

Since the exponential function \(f(x) = a^x\) is defined for all real numbers, the domain is the set of all real numbers.

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Key Concepts

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

function domain
The domain of a function is crucial because it tells us the range of input values (x-values) for which the function is defined and works correctly. It鈥檚 like the set of rules governing what values you can put into the function without breaking it. In the case of exponential functions, these rules are quite broad.

To determine the domain of any function, think about:
  • Are there any values that x cannot take?
  • Will the function ever be undefined for some x value?
For the given exponential function, there are no restrictions on the x values. This means the domain is very flexible and includes all real numbers.

Whether dealing with a simple function or a complex one, identifying the domain is one of the most important initial steps.
exponential function
An exponential function is a mathematical expression where a constant base is raised to a variable exponent. The general form looks like this: \( f(x) = a^x \), where \( a \) is a positive constant and \( x \) is the variable.

Here's why exponential functions are fascinating:
  • They grow rapidly. For instance, even small increases in \( x \) can lead to large increases in \( f(x) \).
  • They are widely used in real-world scenarios, such as population growth, radioactive decay, and interest calculations.
  • They are always positive if \( a > 0 \) and \( x \) is real.
In our specific case, \( f(x) = a^x \) is valid for all real numbers. This function has no restrictions on its input, making it very versatile and widely applicable.
real numbers
Real numbers are the backbone of everyday mathematics. These include both rational numbers (like 2, 0.5, and -3) and irrational numbers (like \( \pi \) and \( \sqrt{2} \)). Essentially, all the numbers on the number line are real numbers.

Let鈥檚 break down their characteristics:
  • They are continuous, meaning there are no gaps.
  • They can be positive, negative, or zero.
  • They can be used to measure quantities, express distances, and calculate anything in real life.
In the context of the problem, knowing that the domain of the exponential function \( f(x) = a^x \) includes all real numbers simplifies our understanding. There are no limitations on \( x \), so any real number can be substituted into the function without causing any issues. This makes working with exponential functions straightforward and broad in application.

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Most popular questions from this chapter

Solve each problem. Global Warming The increasing global temperature can be modeled by the function $$I=0.1 e^{0.02 t}$$ where \(I\) is the increase in global temperature in degrees Celsius since \(1900,\) and \(t\) is the number of years since 1900. a. How much warmer was it in 2010 than it was in \(1950 ?\) b. In what year will the global temperature be \(4^{\circ}\) greater than the global temperature in \(2000 ?\)

Use the following definition. In chemistry, the \(\mathrm{pH}\) of a solution is defined to be $$\mathrm{pH}=-\log \left[H^{+}\right],$$ where \(H^{+}\) is the hydrogen ion concentration of the solution in moles per liter. Distilled water has a pH of approximately 7. A substance with a pH under 7 is called an acid, and one with a pH over 7 is called a base. Tomato juice has a hydrogen ion concentration of \(10^{-4.1}\) moles per liter. Find the \(\mathrm{pH}\) of tomato juice.

Solve each problem. When needed, use 365 days per year and 30 days per month. Present Value Compounding Daily A credit union pays \(6.5 \%\) annual interest compounded daily. What deposit today (present value) would amount to \(\$ 3000\) in 5 years and 4 months? Hint Solve the compound interest formula for \(P\).

Solve \(\log _{2}(x-3)=8.\)

Solve each equation. Find the exact solutions. $$\frac{1}{2^{x}}=5$$
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