The graphs of f and g are given. Find a formula for the function g.

Question

The graphs of f and g are given. Find a formula for the function g.

g(x) = 

 

Answer 1

This can all be very confusing with the different notation, so let's see if we can make it make sense.

To mirror a graph over the x-axis, the y-value has to be opposite. We say g(x) = -f(x). If we had a point of (3,-3) and we wanted it mirrored over the x-axis, it would be (3,3), directly above the first point and equidistant to the x-axis.

To move a graph left or right, we deal with the x-value. Let's use the same (3,-3). If we move it left by 2, our new point is (1,-3), so it seems like g(x) = f(x - 2), right?

Since we are comparing g(x) to f(x), though, it's actually the opposite, because f(x) is 2 more than g(x), so g(x) = f(x + 2). In other words, we use the opposite sign of the direction we are moving left or right. Yeah, strange!

Now, your graphs are not moving up or down, but to do that we again deal with the y-value by using the same sign. To move (3,-3) up to (3, 0), g(x) = f(x) + 3.

To throw that all together, let's mirror f(x) = x² over the x-axis, move it left 2 and up 5. That looks like g(x) = -(x + 2)² + 5.

Also, to mirror over the y-axis, the x-value is opposite. (3,-3) moves to (-3,-3), orrrr g(x) = f(-x).

Long story short, your graph of g(x) is f(x) mirrored over the x-axis and moved left one.

g(x) = -√(x + 1)

Answer 2

Hi Simon, if you suspected that the function f(x) and g(x) looks similar and maybe "related", that's because they are - g(x) is a mirrored and shifted version of f(x).

To see this, let's look at few points on the mirrored version of g(x) ==> -g(x) based on the graph shown

x | -g(x)

-1 | 0

0 | 1

3 | 2

8 | 3

This is essentially f(x) shifted to the left (relative to x=0) by 1, or f(x+1). To see this:

Let's substitute z = x+1

f(0) = sqrt(0) = 0, when z = 0, but since z = x+1 = 0, we have x = -1

f(1) = sqrt(1) = 1, z = 1 --> x+1 = 1 --> x=0

f(4) = sqrt(4) = 2, z = 4 --> x+1 = 4 --> x=3

f(9) = sqrt(9) = 3, z = 9 --> x+1 = 9 --> x=8

But remember we are looking at -g(x), which we found to be:

-g(x) = f(x+1)

g(x) = -f(x+1)

So, g(x) = -sqrt(x+1)

Hope my answer helps.