Question

Use function notation to write g in terms of f

Answer 1

Notice that over the interval [-1 , 1.5], the function f seems to be equal to x^2.

Furthermore, over the interval [-3 , -1], the function f seems to be a straight line of slope equal to 2 and y-intercept equal to 3.

Therefore, we can write down:

`f(x)=\mleft\{\begin{aligned}2x+3\text{ if }-3\leq x<-1 \\ x^2\text{ if }-1\leq x\leq1.5\end{aligned}\mright.`

On the other hand, the function g seems to be a straight line of slope 4 and y-intercept equal to -6 from x=0 to x=2, and something similar to 2x^2 from X=2 to x=4.5, but with its vertex at x=3

`g(x)=\mleft\{\begin{aligned}4x-6\text{ if }0\leq x<2 \\ 2(x-3)^2\text{ if }2\leq x\leq4.5\end{aligned}\mright.`

Notice that the vertical lengths of g seem to be twice those of f, so out first guess may be to write:

`g(x)=2\cdot f(x)`

Additionally, the function seems to be displaced 3 units to the right, so:

`g(x)=2\cdot f(x-3)`

Observe that since the domain of f is equal to [-3 , 1.5], then x has to be in the interval [0 , 4.5] for f(x-3) to be well defined. Also, a change in the correspondence rule of f happens at x=-1, and for g it happens at x=2.

In terms of f, this should happen at x-3=-1, which is equivalent to x=2.

Finally, observe that:

`\begin{gathered} 2\cdot f(x-3)=\mleft\{\begin{aligned}2(2(x-3)+3)\text{ if }-3\leq(x-3)<-1 \\ 2(x-3)^2\text{ if }-1\leq x-3\leq1.5\end{aligned}\mright. \\ =\mleft\{\begin{aligned}2(2x-6+3)\text{ if }-3+3\leq x<-1+3 \\ 2(x-3)^2\text{ if }-1+3\leq x<1.5+3\end{aligned}\mright. \\ =\mleft\{\begin{aligned}4x-6\text{ if }0\leq x<2 \\ 2(x-3)^2\text{ if }2\leq x<4.5\end{aligned}\mright. \\ =g(x) \end{gathered}`

Therefore:

`g(x)=2\cdot f(x-3)`