Question

Which statements represent the relationship between y=4x and y=log4x ?

Select each correct answer.


a. The functions are inverses of each other.

b. The graphs of functions are symmetric to each other over the line y = x.

c. The graphs of functions are symmetric about the line y = 0.

d. The equation ​ y=log4x ​ is the logarithmic form of ​ y=4x ​.

Answer 1

The correct answer is B.

Explanation:

In order to solve this problem, it is recommended to draw the graphs of the functions. By this way, you could easily observe the results.

By giving to
`x` - domain of the function different numbers, you can get the complete picture of each function.

One fact has to be mentioned that the domain of
`y=4^(x)` could be anything (all real numbers). However, the domain of logarithmic function cannot have negative numbers and 0.

When finally we draw the functions of the graphs, we'll get the following result in the picture.

`y=4^(x)` is drawn with the red line, logarithmic function is drawn with the blue line and
`y=x` is drawn with the green line. The latter is used as an answer for the question where it shows that
`y=x` creates a symmetry for the functions.

Answer 2

Option a and b

a. The functions are inverses of each other.

b. The graphs of functions are symmetric to each other over the line y = x.

Explanation:

Given : Equation
`y=4^x` and
`y=\log_4x`

To find : Which statements represent the relationship between two equations?

Solution :

Let,
`f(x)=4^x` and
`g(x)=\log_4x`

First we test whether two functions are inverse or not.

Condition for inverse:
`f(g(x))=g(f(x))=x`

`4^(g(x))=\log_4(f(x))`

`4^(\log_4x)=\log_4(4^x)`

Applying law of logarithm and exponents,

`x=x`

So the two functions are inverses of each other.

The function and its inverse are always symmetrical about the line: y=x

Therefore, Option a and b are correct.

Refer the attached figure below.