Graph f(x) = 4^x and g(x) = log4 x in the same rectangular coordinate system.

Verified step by step guidance

1. Step 1: Understand the functions. The function \( f(x) = 4^x \) is an exponential function, and \( g(x) = \log_4 x \) is a logarithmic function with base 4.
2. Step 2: Identify key points for \( f(x) = 4^x \). Calculate a few values such as \( f(0) = 1 \), \( f(1) = 4 \), and \( f(-1) = \frac{1}{4} \) to understand the behavior of the exponential function.
3. Step 3: Identify key points for \( g(x) = \log_4 x \). Calculate a few values such as \( g(1) = 0 \), \( g(4) = 1 \), and \( g(\frac{1}{4}) = -1 \) to understand the behavior of the logarithmic function.
4. Step 4: Plot the points for both functions on the same coordinate system. For \( f(x) = 4^x \), plot the points (0,1), (1,4), and (-1,0.25). For \( g(x) = \log_4 x \), plot the points (1,0), (4,1), and (0.25,-1).
5. Step 5: Draw smooth curves through the points for each function. The curve for \( f(x) = 4^x \) should rise steeply as \( x \) increases, while the curve for \( g(x) = \log_4 x \) should increase slowly and pass through the origin.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a^x, where 'a' is a positive constant. In this case, f(x) = 4^x represents an exponential growth function, which increases rapidly as 'x' increases. Understanding the properties of exponential functions, such as their domain, range, and asymptotic behavior, is crucial for graphing them accurately.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are expressed as g(x) = log_a(x), where 'a' is the base of the logarithm. For g(x) = log4(x), this function represents the power to which the base 4 must be raised to obtain 'x'. Key characteristics include its domain (x > 0), range (all real numbers), and the fact that it approaches negative infinity as 'x' approaches zero.

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visualize their behavior. When graphing f(x) = 4^x and g(x) = log4(x) together, it's important to recognize their distinct shapes: the exponential function rises steeply, while the logarithmic function increases slowly. Understanding how to find key points, intercepts, and asymptotes will aid in accurately representing both functions on the same graph.