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?

Answer 1

Consider the exponential function $f(x)=2^x.$

By definition, the domain of a function is the set of input argument values for which the function is real and defined.

Let we take $x=2,$ then $f(2)=2^2=4$

$x=4,$ then $f(4)=2^4=16$

$x=10,$ then $f(10)=2^{10}=1024$

If we chose larger values of x, we get larger function values.

For example, If we take $f(0)=2^0=1$

$f(-3)=2^{-3}=\frac{1}{2^3}=\frac{1}{8}$

$f(-10)=2^{-10}=\frac{1}{2^10}=\frac{1}{1024}$

Thus if we choose smaller and smaller values of x. the f unction values will be smaller and smaller functions.

Thus the domain of the function is the set of all real numbers.

Thus the range is limited to the set of positive real numbers. That is, $(0,\infty)$

If we choose larger values of x, we will get larger function values, as the function values will be larger powers of 2.

If we choose smaller and smaller x values, the function values will be smaller and smaller fractions.

Answer 2

Answer:OPTION 4

Step-by-step explanation:

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