Question Find the exponential function passing through $(2,80)$ : $f(x)=4(5)^{x}$ or $f(x)=5(4)^{x}$ ?

Studdy Solution

STEP 1

Assumptions
1. An exponential function has the form $f(x) = ab^x$ where $a$ is the initial value and $b$ is the base of the exponential function.
2. The point through which the function passes is $(2,80)$ , meaning when $x=2$ , $f(x)=80$ .
3. We are given four options and need to determine which one represents an exponential function that satisfies the given point.

STEP 2

We will test each function by substituting $x=2$ and checking if the output is $80$ .

STEP 3

Test the first function $f(x)=4(x)^{5}$ .
$f(2)=4(2)^{5}$

STEP 4

Calculate the value of $f(2)$ for the first function.
$f(2)=4 \cdot 2^5 = 4 \cdot 32 = 128$
Since $f(2) \neq 80$ , the first function is not the correct exponential function.

STEP 5

Test the second function $f(x)=5(x)^{4}$ .
$f(2)=5(2)^{4}$

STEP 6

Calculate the value of $f(2)$ for the second function.
$f(2)=5 \cdot 2^4 = 5 \cdot 16 = 80$
Since $f(2) = 80$ , the second function could be the correct exponential function, but we need to verify that it is indeed exponential. However, upon inspection, we see that the function $f(x)=5(x)^{4}$ is a polynomial function, not an exponential function, because the variable $x$ is in the exponent in an exponential function. Therefore, the second function is not the correct exponential function.

STEP 7

Test the third function $f(x)=4(5)^{x}$ .
$f(2)=4(5)^{2}$

STEP 8

Calculate the value of $f(2)$ for the third function.
$f(2)=4 \cdot 5^2 = 4 \cdot 25 = 100$
Since $f(2) \neq 80$ , the third function is not the correct exponential function.

STEP 9

Test the fourth function $f(x)=5(4)^{x}$ .
$f(2)=5(4)^{2}$

STEP 10

Calculate the value of $f(2)$ for the fourth function.
$f(2)=5 \cdot 4^2 = 5 \cdot 16 = 80$
Since $f(2) = 80$ , the fourth function is the correct exponential function.
The equation that represents an exponential function that passes through the point $(2,80)$ is $f(x)=5(4)^{x}$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms