We are given the graphs of two functions f(x) and g(x) and we are asked to determine the following operations:
[tex](f+g)(x)[/tex]
Part (a): To determine the sum of the two functions we need to have into account the sum of two functions is given by the following relationship:
[tex](f+g)(x)=f(x)+g(x)[/tex]
Since we are required to determine this value at x = -2 we replace "x" in the functions for -2:
[tex](f+g)(-2)=f(-2)+g(-2)[/tex]
Therefore, we need to determine the values of f(-2) and g(-2), we do this using the corresponding graphs. From the graphs we obtained to values:
[tex]\begin{gathered} f(-2)=4 \\ g(-2)=-2 \end{gathered}[/tex]
We obtained them like this:
Now we replace these values and we get:
[tex](f+g)(-2)=4-2=2[/tex]
Therefore, the sum of the functions is 2.
Part (b). We are asked to determine the following:
[tex](f+g)(-1)[/tex]
We use a relationship similar to the previous one:
[tex](f+g)(-1)=f(-1)-g(-1)[/tex]
Now we determine the values of f(-1) and g(-1) from the graph and we get:
[tex]\begin{gathered} f(-1)=1 \\ g(-1)=-3 \end{gathered}[/tex]
In the graph it looks like this:
Now we replace the values and we get:
[tex](f+g)(-1)=1-(-3)=1+3=4[/tex]
Therefore, the difference of the function at x = -1 is 4.
Part (b): We are asked to determine the production of the function at x = 0. We use the following relationship:
[tex](fg)(0)=f(0)g(0)[/tex]
Now we determine the values of the function at x = 0:
[tex]\begin{gathered} f(0)=0 \\ g(0)=-4 \end{gathered}[/tex]
In the graph it looks like this:
Replacing the values we get:
[tex](fg)(0)=(0)(-4)=0[/tex]
Part (d). we are asked to determine the composition of the two functions at x = 0. To do that we use the following relationship:
[tex](g\circ f)(0)=g(f(0))[/tex]
Therefore, we need first to determine the value of f(0) and then evaluate g(x) at that value. The value of f(0) we obtained it in point C and it is:
[tex]f(0)=0[/tex]
Replacing this value we get:
[tex](g\circ f)(0)=g(0)[/tex]
Now we use the value of g(0) that we got in point C:
[tex]g(0)=-4[/tex]
therefore, the composition is:
[tex](g\circ f)(0)=-4[/tex]
Part E: We are asked to determine the quotient between the two functions:
[tex](\frac{f}{g})(-1)=\frac{f(-1)}{g(-1)}[/tex]
We use the values of the functions at x = -1 that we determined in part B. those values are:
[tex]\begin{gathered} f(-1)=1 \\ g(-1)=-3 \end{gathered}[/tex]
Replacing the values:
[tex](\frac{f}{g})(-1)=\frac{1}{-3}=-\frac{1}{3}[/tex]
Therefore, the quotient of the functions is -1/3.
