I am learning calculus and I just could not find the way to reach the answers of two cases
Case 1
If $f(−5)=−2$ and
$g(x)= −2⋅f(x)$, what point
can you determine on the graph of $g$? Answer: $(-5,4)$
Case 2
If $f(−4)=1$ and
$g(x)=f(x/−3)$, what point
can you determine on the graph of $g$? answer: $(12,1)$
What is the process to find these two points?
Thanks a lot, Sonia Ardila.
Best Answer
In the first case, as you know the value of $f$ at the point $x=-5$ (which is $f(-5)=-2$), then you can know $$g(-5)=-2\cdot f(-5) = (-2)\cdot (-2) = 4$$ So $g(-5)=4$ and we know the point $(x,y)=(-5,4)$ belongs to the graph of $g(x)$.
In the second case, as we only know the value of $f(-4)$ (that is $1$), we have to find $x$ such that $f\left(\frac{x}{-3}\right)=f(-4)$ so we can compute $g(x)$. We can make $$\frac{x}{-3}=-4 \Rightarrow x=(-3)\cdot (-4) = 12$$ So we can be sure that $$g(12)=f\left( \frac{12}{-3} \right) = f(-4)=1$$
Therefore the point $(x,y)=(12,1)$ belongs to the graph of $g(x)$.