1.1   Functions

EXERCISE SOLUTIONS

In addition to answering the questions by hand, I also show Maple-assisted solutions.  In red are the actual Maple commands, and in blue the resulting Maple output. 

IMPORTANT:  You do need to be able to perform these calculations using only pencil and paper technology.   If you have any questions about how to solve any of these by hand, please email me. 

Using Maple, this problem can solved like this:

> solve(x=3*y-1,y);

Using Maple, this problem can solved like this:

> solve(x=(y+2)/y,y);

 

Using Maple, this problem can solved like this:

> solve(x=(y-1)/(y+2),y);

Using Maple, this problem can solved like this:

> solve(x=3/(y+1),y);

Now we define these functions we’ll need for the next few problems (to do these problems with Maple)…

> F:=x->3*x-1;

f:=x->(x+2)/x;

g:=x->(x-1)/(x+2);

h:=x->3/(x+1);

Using Maple, this problem can solved like this:

> F(h(x));

> simplify(%);

Here’s this problem worked out “by hand”:

F(x) = 3x  - 1 ,    and so

                     

 

---

Using Maple, this problem can solved like this:

> h(F(x));

 

---

Using Maple, this problem can solved like this:

> g(f(x));

 

> simplify(%);

 

---

8.  By definition of inverse function,   F^–1(F (x)) = x.  

 

---

Using Maple, this problem can solved like this:

This defines "Fi" to be the inverse function of F:

> Fi:=x->solve(F(y)=x,y);

> Fi(x);

It’s easy to check that we’ve really found our desired inverse function:

> F(Fi(x));

> Fi(F(x));

So indeed Fi is the inverse function of F. 

Next let's compute "gi", the inverse function of g:

> gi:=x->solve(g(y)=x,y);

gi(x);

Check to make sure gi is the desired inverse function:

> g(gi(x));  

 

> simplify(%);

> gi(g(x));

Now that we have these inverse functions, we can finally answer this question!   :-)

> gi(Fi(x));

 

---

Using Maple, this problem can solved like this:

> h(h(0));

---

Using Maple, this problem can solved like this:

> h(%);

---

Using Maple, this problem can solved like this:

> g(g(0));

---

Using Maple, this problem can solved like this:

> f(f(1));

---

Using Maple, this problem can solved like this:

> M:=x->2-x^2;

M(M(1));

---

---

Note:   If that was a “–1”  instead of “+1”,
                         then M would be an odd function.
            If that was a “–”  instead of “+” in front of the fg product,
                         then M would be an even function.

---

---

---

---

         Note:   If that was a “–x² ”  instead of “x² ”,
                         then  would be an odd function.

                     If that was a “+ f(x) ”  instead of “– f(x) ” ,
                         then  would be an even function.

---

---