Question

Which of the following exponential functions is represented by the graph? Question 7 options: A) ƒ(x) = (1∕5)x B) ƒ(x) = (1∕5)–x C) ƒ(x) = –5x D) ƒ(x) = 5–x

Answer 1

`f(x) = ((1)/(5))^(-x)`

Explanation:

Given

The graph above

Required

Determine from the list of options, the function of the graph

To determine the function, we have to test each of the options

But first, we nees to write out the points on the graph

(1,5), (0,1), (-1,1/5) and (-2, 1/25)

A.
`f(x) = ((1)/(5))^x`

Point 1: When x = 1, f(x) = 5

`f(x) = ((1)/(5))^x`

`f(1) = ((1)/(5))^1`

`f(1) = ((1)/(5))`

There's no need to check further as the first point doesn't lie on the graph

B.
`f(x) = ((1)/(5))^(-x)`

Point 1: When x = 1, f(x) = 5

`f(x) = ((1)/(5))^(-x)`

`f(1) = ((1)/(5))^(-1)`

`f(1) = 1/((1)/(5))` --- Law of indices

`f(1) = 1*((5)/(1))`

`f(1) = 1*5`

`f(1) = 5`

Point 2: When x = 0, f(x) = 1

`f(0) = ((1)/(5))^(-0)`

`f(0) = ((1)/(5))^(0)`

`f(0) = 1` --- The expression raise to power 0 is 1

Point 3: When x = -1, x = 1/5

`f(x) = ((1)/(5))^(-x)`

`f(-1) = ((1)/(5))^(-(-1))`

`f(-1) = ((1)/(5))^(1)` ----- Law of indices

`f(-1) = ((1)/(5))`

Point 4: When x = -2, x = 1/25

`f(x) = ((1)/(5))^(-x)`

`f(-2) = ((1)/(5))^(-(-2))`

`f(-2) = ((1)/(5))^(2)` ----- Law of indices

`f(-2) = (1)/(5) * (1)/(5)`

`f(-2) = (1)/(25)`

There's no need to check other options as this function satisfy the 4 points on the graph;

Hence;

`f(x) = ((1)/(5))^(-x)` is the exponential function represented on the graph