Question

geraldine is asked to explain the limits on the range of an exponential equation using the function ( f(x) = 2^x ). she makes these two statements:

1. as ( x ) increases infinitely, the ( y )-values are continually doubled for each single increase in ( x ).
2. as ( x ) decreases infinitely, the ( y )-values are continually halved for each single decrease in ( x ).

she concludes that there are no limits within the set of real numbers on the range of this exponential function. which best explains the accuracy of geraldine’s statements and her conclusion?
( \bigcirc ) statement 1 is incorrect because the ( y )-values are increased by 2, not doubled.
( \bigcirc ) statement 2 is incorrect because the ( y )-values are doubled, not halved.
( \bigcirc ) the conclusion is incorrect because the range is limited to the set of integers.
( \bigcirc ) the conclusion is incorrect because the range is limited to the set of positive real numbers.

Explanation:

Step1: Analyze the exponential function \( f(x) = 2^x \)

For an exponential function \( a^x \) (where \( a>0, a
eq1 \)), the range is the set of positive real numbers. For \( f(x)=2^x \):

• When \( x \) increases, \( 2^x \) grows exponentially (each increase in \( x \) by 1 multiplies the \( y \)-value by 2, so statement 1 is correct in the doubling aspect as \( x \) increases).
• When \( x \) decreases (e.g., \( x \) becomes \( x - 1 \)), \( 2^{x-1}=\frac{2^x}{2} \), so the \( y \)-value is halved, so statement 2 is correct.

Step2: Analyze the conclusion

The range of \( f(x)=2^x \) is \( (0, +\infty) \), which is the set of positive real numbers (not all real numbers, since \( 2^x>0 \) for all real \( x \)). So Geraldine's conclusion that there are no limits within the set of real numbers on the range is incorrect because the range is limited to positive real numbers.

Now let's analyze each option:

• Option 1: Statement 1 is correct (as \( x \) increases, \( y \)-values double for each \( x \) increase), so this is wrong.
• Option 2: Statement 2 is correct (as \( x \) decreases, \( y \)-values halve), so this is wrong.
• Option 3: The range is not limited to integers (e.g., \( 2^{0.5}=\sqrt{2} \) is not an integer), so this is wrong.
• Option 4: The range of \( 2^x \) is \( y>0 \) (positive real numbers), so the conclusion is incorrect because the range is limited to positive real numbers. This is correct.

Answer:

The conclusion is incorrect because the range is limited to the set of positive real numbers. (The corresponding option is: The conclusion is incorrect because the range is limited to the set of positive real numbers.)