Question

snc.algebra iii - standard (10/29)exponential growth functionsgeraldine is asked to explain the limits on the range of an exponential equation using the function $f(x)=2^x$. she makes two statements.1. as a increases infinitely, the y-values are consistently doubled for each single increase in x.2. as a decreases infinitely, the y-values are consistently 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?the conclusion is incorrect because the range is limited to the set of positive real numbers.statement 1 is incorrect because the y-values are increased by 2, not doubled.statement 2 is incorrect because the y-values are doubled, not halved.the conclusion is incorrect because the range is limited to the set of integers

Explanation:

Step1: Analyze the function

The given exponential function is $f(x)=2^x$. For all real $x$, $2^x > 0$, so the range is positive real numbers, not all real numbers.

Step2: Evaluate Statement 1

For $f(x)=2^x$, when $x$ increases by 1: $f(x+1)=2^{x+1}=2\times2^x=2f(x)$. So $y$-values double, meaning Statement 1's claim that they are "increased by 2, not doubled" is wrong.

Step3: Evaluate Statement 2

When $x$ decreases by 1: $f(x-1)=2^{x-1}=\frac{2^x}{2}=\frac{f(x)}{2}$. So $y$-values are halved, meaning Statement 2's claim that they are "doubled, not halved" is wrong.

Step4: Evaluate conclusions

Geraldine's conclusion that the range has no limits in all real numbers is incorrect because the range is only positive real numbers.

Answer:

1. Statement 1 is incorrect because the y-values are increased by 2, not doubled
2. Statement 2 is incorrect because the y-values are doubled, not halved
3. The conclusion is incorrect because the range is limited to the set of positive real numbers