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Which properties are present in a table that represents an exponential function in the form [tex]y = b^x[/tex] when [tex]b \ \textgreater \ 1[/tex]?

1. As the [tex]x[/tex]-values increase, the [tex]y[/tex]-values increase.
2. The point [tex](1, 0)[/tex] exists in the table.
3. As the [tex]x[/tex]-values increase, the [tex]y[/tex]-values decrease.
4. As the [tex]x[/tex]-values decrease, the [tex]y[/tex]-values decrease, approaching a singular value.

A. I and IV
B. I and II
C. II and III
D. III only


Sagot :

To determine which properties are present in a table that represents an exponential function in the form \(y=b^x\) where \(b>1\), let's analyze each statement in detail:

Property I: As the \(x\)-values increase, the \(y\)-values increase.

For an exponential function \(y = b^x\) with \(b > 1\):
- When \(x\) increases, \(b^x\) also increases because the base \(b\) raised to a higher power results in a larger number.
- Therefore, this property is valid.

Property II: The point \((1,0)\) exists in the table.

For an exponential function \(y = b^x\) with \(b > 1\):
- If \(x = 1\), then \(y = b^1 = b\), not 0.
- Therefore, this property is invalid because for any \(b > 1\), \(y\) will never be 0 when \(x = 1\).

Property III: As the \(x\)-values increase, the \(y\)-values decrease.

For an exponential function \(y = b^x\) with \(b > 1\):
- When \(x\) increases, \(y = b^x\) increases as stated before.
- Therefore, this property is invalid because \(y\) does not decrease with increasing \(x\).

Property IV: As the \(x\)-values decrease, the \(y\)-values decrease, approaching a singular value.

For an exponential function \(y = b^x\) with \(b > 1\):
- When \(x\) decreases, \(y = b^x\) decreases and approaches 0 as \(x\) approaches negative infinity.
- Therefore, this property is valid as \(y\) gets closer and closer to 0.

Thus, the valid properties for an exponential function \(y = b^x\) where \(b>1\) are:

- Property I: As the \(x\)-values increase, the \(y\)-values increase.
- Property IV: As the \(x\)-values decrease, the \(y\)-values decrease, approaching a singular value (0).

Therefore, the correct answer is:

I and IV