Answer:
f(x) is a one to one function.That maps out one function to another ,and it therefore passes both the vertical and horizontal I ine test
Answer with explanation:
The given linear function is
f(x)= 2 x -3
y=2 x -3
[tex]y+3=2x\\\\x=\frac{y+3}{2}[/tex]
Replacing , x by y, and y by x, we will get the Inverse of the given function
[tex]y=\frac{x+3}{2}[/tex]
Since ,for every x, there is a unique y, or for every, y there is unique x, therefore it is a function.
We can check this also by using the concept of function, that is the function must be one one and Onto.
To check whether, the linear function is one- one,
If ,f(a)=f(b), then , a=b.
f(a)= 2 a -3
f(b)=2 b -3
2 a -3 = 2 b -3
2 a= 2 b
a=b
So, f(x)=2 x -3, is one -one.
To Check whether it is onto
for, f(x)=y
there must be unique x, for unique y
[tex]y=\frac{x+3}{2}[/tex]
So , f(x) is one one and onto.Which shows that f(x) has an Inverse.
Otherwise , we will use Horizontal line test.
Draw the graph of , f(x)=2 x -3
Horizontal lines does not cut the function, more than once.
or, you can find the inverse of f(x),then check whether it passes Horizontal line test.That is, function , 2 y = x +3, the inverse function Passes Horizontal Line test.
Option C: The graph of the inverse of f(x) passes the horizontal line test.
