In the graph, two functions are depicted: $f(x) = 2^x$ and $g(x) = \log_2(x)$, along with the line $y = x$. Let's analyze the domain and range of each function.

1. Function $f(x) = 2^x$:

This is an exponential function where the base is 2. The general form of an exponential function $a^x$, where $a > 1$, shows rapid growth as $x$ increases.

• Domain: The domain of $f(x) = 2^x$ consists of all real numbers, since you can raise 2 to any real exponent without restriction. $\text{Domain of } f(x): (-\infty, \infty)$

• Range: The output values of an exponential function are always positive because $2^x$ is always greater than 0 for any real $x$. Therefore, the range is all positive real numbers. $\text{Range of } f(x): (0, \infty)$

2. Function $g(x) = \log_2(x)$:

This is a logarithmic function with base 2. The logarithmic function is the inverse of the exponential function, which means it "undoes" what the exponential function does.

• Domain: The logarithm of a number is only defined for positive values of $x$ (i.e., $x > 0$). Therefore, the domain of $g(x) = \log_2(x)$ is restricted to positive real numbers. $\text{Domain of } g(x): (0, \infty)$

• Range: The range of a logarithmic function is all real numbers. For $g(x) = \log_2(x)$, as $x \to 0^+$, $g(x) \to -\infty$, and as $x \to \infty$, $g(x) \to \infty$. Therefore, the range covers all real values. $\text{Range of } g(x): (-\infty, \infty)$

Summary of Domain and Range:

• $f(x) = 2^x$:

  • Domain: $(-\infty, \infty)$
  • Range: $(0, \infty)$

• $g(x) = \log_2(x)$:

  • Domain: $(0, \infty)$
  • Range: $(-\infty, \infty)$

The line $y = x$ shown in the graph represents the fact that these two functions are inverses of each other, which is why $f(x) = 2^x$ is reflected over the line $y = x$ to become $g(x) = \log_2(x)$.

Would you like any further clarifications or details?

Here are 5 related questions:

1. How does the base of the exponential or logarithmic function affect its graph?
2. What happens to the domain and range if we change the base of the logarithmic function?
3. Can you explain the relationship between exponential and logarithmic functions in terms of their inverses?
4. How does the graph of $f(x) = 2^x$ change if the base is less than 1, for example, $f(x) = (1/2)^x$?
5. How does transforming the graph, such as shifting or stretching, affect the domain and range of these functions?

Tip: For exponential functions, the base affects the rate of growth or decay. A base greater than 1 results in growth, while a base between 0 and 1 results in decay.