Question

Which function has an inverse that is also a function? g(x) = 2x – 3 k(x) = –9x2 f(x) = |x + 2| w(x) = –20

Answer 1

The inverse of g(x) is also a function because g(x) is one - one .

The functions f(x) , k(x), w(x) are not one -one functions because more than one element have same images.

Then their inverse functions will assign more than one image for each element.

Hence inverse of these functions are not functions.

Answer 2

Final Answer : g(x) is invertible function.

Steps and Understanding :
1) A function is invertible iff it is one-one and onto.
For one -one ,
If there exists a line parallel to x-axis which cuts the graph of the function at least two points then function is many -one other wise one-one.


2).We look at all graphs of given functions as shown in pic.
$pic = 1 \: \: \: g(x) = 2x - 3 \\ \\ pic = 2 \: \: \: k(x) = - 9 {x}^{2} \\ \\ pic = 3 \: \: \: f(x) = |x + 2| \\ \\ pic = 4 \: \: \: w(x) = - 20$


3) For - onto function :
Co-domain = Range.
If Co-domain is not given, then it it is assumed as set of all real numbers.

4) Now, by graph
we observe that
k(x), f(x) ,w(x) are many one function.
So, they are not invertible function.


5) Remaining is g(x) which is one -one.
And g(x) is onto function too as Range of g(x) is Set of all real numbers.
So,
Finally, g(x) is an invertible function.