Question

identifying characteristics of the exponential function \\(y = b^x\\) (\\(b > 1\\))

the domain of an exponential function is all real numbers. the range of an exponential function is
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• all real numbers
• nonnegative real numbers
• positive real numbers
• negative real numbers

Explanation:

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<reasoning>

Analyze the graph of the exponential function

The graph shows a standard exponential growth function \(y = b^x\) where \(b > 1\).
As \(x\) approaches negative infinity, the curve approaches but never touches the x-axis (\(y = 0\)), indicating a horizontal asymptote at \(y = 0\).
As \(x\) increases, the curve rises rapidly towards positive infinity.

Determine the range

Since the graph is strictly above the x-axis and never reaches \(y = 0\), the output values \(y\) must be strictly greater than zero.
Therefore, the range of the exponential function is all positive real numbers.
</reasoning>

<answer>
The domain of an exponential function is <blank>all real numbers</blank>. The range of an exponential function is <blank>positive real numbers</blank>.
</answer>

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Answer:

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<reasoning>

Analyze the graph of the exponential function

The graph shows a standard exponential growth function \(y = b^x\) where \(b > 1\).
As \(x\) approaches negative infinity, the curve approaches but never touches the x-axis (\(y = 0\)), indicating a horizontal asymptote at \(y = 0\).
As \(x\) increases, the curve rises rapidly towards positive infinity.

Determine the range

Since the graph is strictly above the x-axis and never reaches \(y = 0\), the output values \(y\) must be strictly greater than zero.
Therefore, the range of the exponential function is all positive real numbers.
</reasoning>

<answer>
The domain of an exponential function is <blank>all real numbers</blank>. The range of an exponential function is <blank>positive real numbers</blank>.
</answer>

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