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 geraldines 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:

1. Analyze Statement 1: For \( f(x) = 2^x \), when \( x \) increases by 1 (e.g., from \( x = 1 \) to \( x = 2 \)), \( f(1)=2^1 = 2 \), \( f(2)=2^2 = 4 \), so \( y \)-values double for each single increase in \( x \). As \( x \to \infty \), \( 2^x \to \infty \), so Statement 1 is correct.
2. Analyze Statement 2: When \( x \) decreases by 1 (e.g., from \( x = 1 \) to \( x = 0 \)), \( f(1)=2 \), \( f(0)=1 \), so \( y \)-values are halved. As \( x \to -\infty \), \( 2^x=\frac{1}{2^{|x|}}\to 0 \), so Statement 2 is correct.
3. Analyze the conclusion: The range of \( f(x)=2^x \) is \( (0, \infty) \) (positive real numbers), not all real numbers. So the conclusion that there are no limits on the range within real numbers is incorrect because the range is limited to positive real numbers.

• Check other options:
• First option: Statement 1 is correct, so this is wrong.
• Second option: Statement 2 is correct, so this is wrong.
• Third option: The range is not limited to integers (e.g., \( 2^{0.5}=\sqrt{2} \) is in the range), so this is wrong.
• Fourth option: The range of \( 2^x \) is positive real numbers, so the conclusion is incorrect as it claims no limits in real numbers.

Answer:

The conclusion is incorrect because the range is limited to the set of positive real numbers.