Question

What is the value of k ? It seems easy but what are the real steps ?

Answer 1

Looks like f(x) = 2x + 3 ...

Explanation:

...just by inspection.

`f^-1(x) = (x - 3)/2`

and then `f^-1(7) = 2` (which you can also read directly from diagram 6)

and

`k = f^-1(11) = (11-3)/2 = 4`

Answer 2

a) `f^(-1)(7)` can be read directly from the supplied diagram:
`color(white)("XXX")f^(-1)(7)=2`

b)
We are told that the function is linear.
Therefore
`color(white)("XXX")f(x)=color(green)mx+color(blue)c`
for some constants `color(green)m` and color(blue)c#

We know from the diagram that:
`color(white)("XXX")f(color(magenta)0)=color(green)m * color(magenta)0+color(blue)c=3`
`color(white)("XXX")rarr color(blue)c=color(blue)3`
also
`color(white)("XXX")f(color(magenta)2)=color(green)m * color(magenta)2 +color(blue)c= 7`
`color(white)("XXX")`and since `color(blue)c=color(blue)3`
`color(white)("XXXXXX")color(magenta)2color(green)m=4` and
`color(white)("XXXXXX")color(green)m=2`

So the function is
`color(white)("XXX")f(x)=color(green)2x+color(blue)3`

If `f(x)=11`
we have
`color(white)("XXX")color(green)2x+color(blue)3=11`
`color(white)("XXX")rarr color(green)2x=8`
`color(white)("XXX")rarr x=4`