Question

Which of the following represents Exponential Growth? Choose all that apply.

Group of answer choices
y=(5/4)^x

y=4^-x

y=5^x

y=(4/5)^x

Answer 1

The functions y=(5/4)^x and y=5^x represent exponential growth since the value of the function increases as the exponent (x) increases. This pattern follows the characteristic of exponential growth, where larger values lead to faster rates of increase.

Step-by-step explanation:

Exponential growth occurs in mathematics when a quantity increases by a fixed percentage or fraction over equal intervals of time. This can be represented by a mathematical function where the variable is an exponent. Looking at the provided options, let's determine which ones represent exponential growth:

• y=(5/4)^x: This function represents exponential growth because as x increases, the value of the function also increases proportionally.
• y=4^-x: This function does not represent exponential growth; instead, it represents exponential decay, as the value of the function decreases as x increases.
• y=5^x: This is a classic example of exponential growth, with the base 5 being raised to the power of x. As x increases, the function's value grows exponentially.
• y=(4/5)^x: Similar to the second function, this represents exponential decay, not growth, as the function's value diminishes as x increases.

Therefore, the functions that represent exponential growth are y=(5/4)^x and y=5^x.

Answer 2

`y=((5)/(4))^x`

`y=5^x`

Step-by-step explanation:

we know that

In a exponential function of the form

`y=a(b^x)`

where

a is the initial value

b is the base

If the value of b >1 ----> then is a exponential growth

If the value of b <1 ----> then is a exponential decay

Verify each case

case a) we have

`y=((5)/(4))^x`

`b=(5)/(4)`

b> 1

therefore

The function represent exponential growth

case b) we have

`y=(4)^(-x)`

`b=(1)/(4)`

b< 1

therefore

The function represent exponential decay

case c) we have

`y=5^x`

`b=5`

b> 1

therefore

The function represent exponential growth

case d) we have

`y=((4)/(5))^x`

`b=(4)/(5)`

b< 1

therefore

The function represent exponential decay