Question

graphing exponential functions
instruction active
identifying characteristics of the exponential function $y = b^x$ ($b > 1$)
the domain of an exponential function is all real numbers (dropdown selected). the range of an exponential function is dropdown to fill.
done (button)
(graph of exponential function $y = b^x$ ($b > 1$) with x-axis and y-axis, curve increasing, passing near (0,1) and approaching x-axis as x→-∞)

Explanation:

Step1: Recall Exponential Function Properties

For the exponential function \( y = b^x \) where \( b>1 \), we analyze its range. The exponential function \( b^x \) (with \( b > 0, b
eq1 \)) has the property that when \( x \) approaches \( -\infty \), \( b^x \) approaches \( 0 \) (since \( b>1 \), \( b^{-\infty}=\frac{1}{b^{\infty}} = 0 \)), and as \( x \) approaches \( \infty \), \( b^x \) approaches \( \infty \). Also, for all real \( x \), \( b^x>0 \) because any positive number raised to a real power is positive.

Step2: Determine the Range

From the graph, we can see that the curve is above the \( x \)-axis ( \( y = 0 \) ) and goes to infinity as \( x \) increases. So the range of \( y=b^x \) ( \( b > 1 \) ) is all positive real numbers, or in interval notation \( (0, \infty) \).

Answer:

all positive real numbers (or \( y > 0 \) or \( (0, \infty) \))