Question

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

the graph of an exponential function has a y-intercept only.

the graph is increasing throughout its domain and has at the x-axis because the values get close to 0 but never reach it.
options for the dropdown:

• a slope line
• an asymptote
• a tangent line

Explanation:

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<reasoning>

Analyze the domain and range

Using the Domain of Exponential Functions and Range of Exponential Functions knowledge points

• The domain of the parent exponential function \(y = b^x\) (\(b > 1\)) consists of all real numbers, as \(x\) can take any real value.
• The range consists of all positive real numbers, as \(b^x > 0\) for all real \(x\).

Analyze the intercepts

Using the Exponential Function Graphs knowledge point

• Since \(b^x > 0\), there is no \(x\)-intercept.
• When \(x = 0\), \(y = b^0 = 1\), giving a \(y\)-intercept at \((0, 1)\). Thus, the graph has a \(y\)-intercept only.

Analyze the behavior and asymptote

Using the Exponential Function Graphs knowledge point

• Since \(b > 1\), the function values increase as \(x\) increases, meaning the graph is increasing throughout its domain.
• As \(x \to -\infty\), the values of \(b^x\) approach \(0\) but never reach \(0\). This horizontal boundary line \(y = 0\) (the \(x\)-axis) is called an asymptote.

</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>.

The graph of an exponential function has <blank>a y-intercept only</blank>.

The graph is <blank>increasing</blank> throughout its domain and has <blank>an asymptote</blank> at the x-axis because the values get close to 0 but never reach it.
</answer>

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Answer:

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<reasoning>

Analyze the domain and range

Using the Domain of Exponential Functions and Range of Exponential Functions knowledge points

• The domain of the parent exponential function \(y = b^x\) (\(b > 1\)) consists of all real numbers, as \(x\) can take any real value.
• The range consists of all positive real numbers, as \(b^x > 0\) for all real \(x\).

Analyze the intercepts

Using the Exponential Function Graphs knowledge point

• Since \(b^x > 0\), there is no \(x\)-intercept.
• When \(x = 0\), \(y = b^0 = 1\), giving a \(y\)-intercept at \((0, 1)\). Thus, the graph has a \(y\)-intercept only.

Analyze the behavior and asymptote

Using the Exponential Function Graphs knowledge point

• Since \(b > 1\), the function values increase as \(x\) increases, meaning the graph is increasing throughout its domain.
• As \(x \to -\infty\), the values of \(b^x\) approach \(0\) but never reach \(0\). This horizontal boundary line \(y = 0\) (the \(x\)-axis) is called an asymptote.

</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>.

The graph of an exponential function has <blank>a y-intercept only</blank>.

The graph is <blank>increasing</blank> throughout its domain and has <blank>an asymptote</blank> at the x-axis because the values get close to 0 but never reach it.
</answer>

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