# Graph the function $g(x) = 3f(({x\over2}) − 1) − 3$

I am looking at the problem set from MIT 18.01 (single variable calculus), in the first problem set I am stuck on this function.

The solution to this function is given as:

The graph is made up of segments joining (0, −6) to (4, 3) to (8, −6). It repeats in a zigzag with period 8. This can be derived using:

(1) $${x\over2} − 1 = −1 =⇒ x = 0$$ and $$g(0) = 3f(−1) − 3 = −6$$

(2) $${x\over2} − 1 = 1 =⇒ x = 4$$ and $$g(4) = 3f(1) − 3 = 3$$

(3) $${x\over2} − 1 = 3 =⇒ x = 8$$ and $$g(8) = 3f(3) − 3 = −6$$

I have understood how the value of x has been derived but can't figure out the respective values at g(0), g(4), g(8).

Could someone tell me how the g(x) or y values have been calculated?

• What is $f(x)$? – pwerth Feb 1 at 22:13
• F, according to an earlier part of this question, is a piecewise linear function, consists of straight line segments joining the points: (-1,-1), (1,2), (3,-1) and (5,2). – Noel Feb 1 at 23:00
• You can immediately solve equations 1, 2 and 3 for the values of $f(-1),f(1)$ and $f(3)$. – John Wayland Bales Feb 1 at 23:04
• I only could understand how the value of x was found, the value for g(x) where x={0,4,8} has been eluding me, i know the solution is simple, but i am having a hard time seeing it for the g(x) part of the function – Noel Feb 1 at 23:10
• When you say "The graph is made up of segments joining . . ." do you mean the graph of $g$? If that is the case then one could guess that you want to find the graph of $f$ (since you do not say what you want to find). Do you want to find the graph of $f$? The equation of $f$? What exactly do you want to find? – John Wayland Bales Feb 1 at 23:12