Question

Which of the following functions are one-to-one? Select all that apply.

Answer 1

1,2 and 5 is the answers I think

Answer 2

`f(x)=x^3-7\,,\,f(x)=(1)/(8x-1)` are one-to-one

Explanation:

A function y = f(x) is said to be one-to-one if
`f(x_1)=f(x_2)\Rightarrow x_1=x_2`

`f(x)=x^3-7`

`f(x_1)=f(x_2)\\x_1^3-7=x_2^3-7\\x_1^3=x_2^3\\x_1=x_2`

So, f is one-to-one.

`f(x)=x^2-4`

`f(1)=1^2-4=1-4=-3\\f(-1)=(-1)^2-4=1-4=-3\\\Rightarrow f(1)=f(-1)\\\text{but}\,\,1\\eq -1`

So, f is not one-to-one

`f(x)=(1)/(8x-1)`

`f(x_1)=f(x_2)\\(1)/(8x_1-1)=(1)/(8x_2-1)\\8x_1-1=8x_2-1\\8x_1=8x_2\\x_1=x_2`

So, f is one-to-one

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

`f(1)=(5)/(1^4)=5\\f(-1)=(5)/((-1)^4)=5\\f(1)=f(-1)\,\,but\\,\,1\\eq -1`

So, f is not one-to-one

`f(x)=\left | x \right |`

`f(1)=\left | 1 \right |=1\\f(-1)=\left | -1 \right |=1\\f(1)=f(-1)\,\,but\,\,1\\eq -1`

So, f is not one-to-one

Therefore, functions
`f(x)=x^3-7\,,\,f(x)=(1)/(8x-1)` are one-to-one