Example 3.1.1
Use the horizontal line test to show that the function  is not one-to-one.

Of course we make a sketch.

>	f:=x->x^2+3*x+7;

>	plot(f(x), x=-4..3);

Clearly there are many horizontal lines which intersect with this graph more than once.    is an example as can be seen in the next graph.

>	plot({f(x), 9}, x=-4..3);

The function f is therefore not one-to-one.

Example 3.1.2
Use the definition of a one-to-one function to show that  is one-to-one.

We will show that  implies that .

>	f:=x->(3*x+2)/(x-7);

>	eq:=f(x[1])=f(x[2]);

>	test:=solve(eq, {x[1]});

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Example 3.2.1
Compute the inverse function of .

We code the equation  and solve for x.

>	f:=x->5*x/(4-3*x);

>	eq:=y=f(x);

>	finv_y:=solve(eq, x);

>	finv:=subs(y=x, finv_y);

This Maple  expression can be converted into a Maple  function by using the unapply command.

>	finv:=unapply(finv, x);

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Example 3.2.2
Consider the function , 5 x 8.
i)     Show that   is one-to-one.
ii)    Find the inverse of .
iii)   Sketch  and its inverse in one picture

i)   We use the horizontal line test.

>	f:=x->sqrt(x-5);

>	plot(f(x), x=5..8, color=blue);

>

Clearly no horizontal line intesects with this graph more than once, the function is therefore one-to-one.

ii)  The inverse of  can be found in the usual way.

>	eq:=y=f(x);

>	finv_y:=solve(eq, x);

>	finv:=subs(y=x, finv_y);

>	finv:=unapply(finv, x);

>

iii)  When it comes to plotting the functions  and   we have to keep in mind that they have different domains.  Clearly the function  has [5, 8] as its domain, but the domain of  equals the range of  and is therefore given by .  In order to handle these different domains we create two separate graphic images, one for  and the other for .

>	p1:=plot(f(x), x=5..8):

>	p2:=plot(finv(x), x=f(5)..f(8), color=blue):

Observe that these statements end with a colon rather than a semicolon.  This prevents premature display of the output.

We now use the display  command to simultaneously show the graphs of  and .  Observe that if we choose equal scales on the coordinate axes, the two graphs are each others mirror image in the line y=x.

>	display([p1, p2], scaling=constrained);

This mirror image property of graphs of inverse functions allows us to quickly sketch the graph of a one-to-one function and its inverse without actually computing that inverse.  To do this we make use of  a parametric plot.

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Example 3.2.3
Sketch in one picture the graph of the one-to-one function  and its inverse.

We use Maple's parametric plot routine and to ensure equal scaling on the coordinate axes we include the scaling=constrained  option.

>	f:=x->x^3+2;

>	plot({[t, f(t), t=-1.3..1], [f(t), t, t=-1.3..1]}, scaling=constrained);

Of course the line  can easily be added to the picture.

>	plot([[t, f(t), t=-1.3..1], [f(t), t, t=-1.3..1], [t, t, t=-1..3]], color=[red, green, blue], scaling=constrained);

>