The complete graph of $y=f(x),$ which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1.$) Let $a$ and $b$ be the largest negative integer and the smallest positive integer, respectively, such that the functions $g(x)=f(x)+ax$ and $h(x)=f(x)+bx$ are invertible. What is $a^2+b^2?$

I think someone posted this question before, but it wasn't answered. I don't know how to draw the graph and he has it, the link for the graph is https://web2.0calc.com/questions/the-complete-graph-of-y-f-x-which-consists-of-five

Anyway thanks!

EvanWei123 Jul 30, 2019

#4**+3 **

For a function to be invertible, it must pass the "horizontal line test," which means you must not be able to draw a horizontal line that intersects the graph more than once. We can see the graph of y = f(x) by itself does not pass the horizontal line test and so is not invertible.

Let's imagine what g(x) would look like if a = 1

g(x) = f(x) + ax If a = 1 then

g(x) = f(x) + x

Let's see what g(1) would be.

g(1) = f(1) + 1 And from the picture we can see f(1) = -5

g(1) = -5 + 1

g(1) = -4 So the point (1, -4) is on the graph of g(x)

Adding the x-value to the y-value ended up shifting that point up.

Let's see what g(3) would be.

g(3) = f(3) + 3 And from the picture we can see f(3) = 2

g(3) = 2 + 3

g(3) = 5 So the point (3, 5) is on the graph of g(x)

Adding the x-value to the y-value ended up shifting that point up (even more than last time).

Let's see what g(-1) would be.

g(-1) = f(-1) - 1 And from the picture we can see f(-1) = 3

g(-1) = 3 - 1

g(-1) = 2 So the point (-1, 2) is on the graph of g(x)

Adding the x-value to the y-value ended up shifting the point down.

Let's see what g(-2) would be.

g(-2) = f(-2) - 2 And from the picture we can see f(-2) = 5

g(-2) = 5 - 2

g(-2) = 3 So the point (-2, 3) is on the graph of g(x)

Adding the x-value to the y-value ended up shifting that point down (even more than last time).

And if we increase the value of a , that will only exaggerate the changes.

By now we can see what g(x) should look like compared to f(x)

Here is a graph of f(x) and g(x) with a slider for different values of a:

https://www.desmos.com/calculator/eoqgwwn620

Moving the slider to the right until the graph "straightens out" enough to pass the horizontal line test, we can see that the smallest positive integer that makes g(x) invertible is 5. Moving the slider to the left (and zooming out as necessary), we can see that the largest negative integer that makes g(x) invertible is -4.

(-4)^{2} + 5^{2} = 16 + 25 = 41

hectictar Jul 31, 2019

#5**+3 **

Hectictar thank you so much! Sorry if i wasted your time. Your solution is clear, smart, and correct!

EvanWei123 Aug 2, 2019