Answer:
True. The domain of the function f(x) = a^x, where a > 0 and a cannot equal 1 is the set of all real numbers. This means that any real number can be used as the input (x) for this function. The domain is the set of all possible inputs for a function. The function f(x) = a^x is an exponential function, which has the form y = b^x, where b is the base and x is the exponent. The base b must be positive and not equal to 1, otherwise the function would be constant or undefined. The exponent x can be any real number, such as integers, fractions, decimals, irrational numbers, etc. Therefore, the domain of f(x) = a^x is (-∞, ∞), which means all real numbers from negative infinity to positive infinity . Here is a graph of the function f(x) = 2^x as an example:
Python
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# Import matplotlib.pyplot for plotting
import matplotlib.pyplot as plt
# Define the function f(x) = 2^x
def f(x):
return 2**x
# Create a list of x values from -5 to 5
x_values = [x for x in range(-5, 6)]
# Create a list of y values by applying the function to each x value
y_values = [f(x) for x in x_values]
# Plot the points (x, y) using a red line
plt.plot(x_values, y_values, 'r-')
# Label the x-axis and y-axis
plt.xlabel('x')
plt.ylabel('f(x)')
# Show the graph
plt.show()
Output:
!A graph of the function f(x) = 2^x with a red curve that starts from the bottom left and rises to the top right. The x-axis is labeled as x and the y-axis is labeled as f(x). The graph shows that the function is defined for all real values of x.