Chapter 4: Define and Describe 

`*`(Math, `*`(Problem, `*`(Set)));  

1.      Given  determine  

f := proc (x) options operator, arrow, function_assign; `/`(`*`(`+`(`*`(4, `*`(x)), `-`(1))), `*`(`+`(`*`(2, `*`(x)), 3))) end proc;  

f := proc (x) options operator, arrow; `/`(`*`(`+`(`*`(4, `*`(x)), `-`(1))), `*`(`+`(`*`(2, `*`(x)), 3))) end proc; (4.1.1)
 

solve(f(y) = x, y);  

`+`(`-`(`*`(`+`(`*`(3, `*`(x)), 1), `*`(`/`(`+`(`*`(2, `*`(x)), `-`(4))))))); (4.1.2)
 

 

finverse(x); (4.1.3)
 

finverse(-5);  

-1; (4.1.4)
 

As a check: 

f(-1);  

-5; (4.1.5)
 

 

 

 

 

 

2.       Assume `and`(`<=`(0, theta), `<=`(theta, `+`(`*`(`/`(1, 2), `*`(Pi))))); .  Given theta = arctan(`/`(5, 12)); , determine  

theta := arctan(`/`(5, 12));  

theta := arctan(`/`(5, 12)); (4.2.1)
 

sin(theta);  

`/`(5, 13); (4.2.2)
 

OR 

sin(arctan(`/`(5, 12)));  

`/`(5, 13); (4.2.3)
 

 

 

3.      Assume `and`(`<=`(0, theta), `<=`(theta, `+`(`*`(`/`(1, 2), `*`(Pi))))); .  Given theta = arccos(`/`(`*`(sqrt(x)), `*`(`+`(x, 1)))); , determine tan(theta); and  

 

restart;  

theta := arccos(`/`(`*`(sqrt(x)), `*`(`+`(x, 1))));  

theta := arccos(`/`(`*`(sqrt(x)), `*`(`+`(x, 1)))); (4.3.1)
 

tan(theta);  

`/`(`*`(sqrt(`+`(1, `-`(`/`(`*`(x), `*`(`^`(`+`(x, 1), 2)))))), `*`(`+`(x, 1))), `*`(sqrt(x))); (4.3.2)
 

 

`/`(`*`(sqrt(`+`(`*`(`^`(x, 2)), x, 1))), `*`(sqrt(x))); (4.3.3)
 

tan(`+`(`*`(2, `*`(theta))));  

tan(`+`(`*`(2, `*`(arccos(`/`(`*`(sqrt(x)), `*`(`+`(x, 1)))))))); (4.3.4)
 

 

`+`(`/`(`*`(2, `*`(sqrt(`+`(1, `-`(`/`(`*`(x), `*`(`^`(`+`(x, 1), 2)))))), `*`(`+`(x, 1)))), `*`(sqrt(x), `*`(`+`(1, `-`(`/`(`*`(`+`(1, `-`(`/`(`*`(x), `*`(`^`(`+`(x, 1), 2))))), `*`(`^`(`+`(x, 1), 2)... (4.3.5)
 

 

`+`(`-`(`/`(`*`(2, `*`(sqrt(`+`(`*`(`^`(x, 2)), x, 1)), `*`(sqrt(x)))), `*`(`+`(`*`(`^`(x, 2)), 1))))); (4.3.6)
 

 

4.       Given Bob(x) = `^`(2, x);  and Milly(x) = `+`(`*`(`^`(x, 2)), 1);   , determine: 

 

`^`(2, x); Bob(x); `+`(`*`(`^`(x, 2)), 1); Milly(x);  

`/`(`*`(Milly(x)), `*`(Bob(x)));  

`/`(`*`(`+`(`*`(`^`(x, 2)), 1)), `*`(`^`(2, x))); (4.4.1)
 

sqrt(Milly(x));  

sqrt(`+`(`*`(`^`(x, 2)), 1)); (4.4.2)
 

(`@`(Bob, Milly))(x);  

`^`(2, `+`(`*`(`^`(x, 2)), 1)); (4.4.3)
 

solve(Bob(y) = x, y);  

`/`(`*`(ln(x)), `*`(ln(2))); (4.4.4)
 

 

BobInv(x); (4.4.5)
 

`+`(BobInv(2), `/`(1, `*`(Milly(4))));  

`/`(18, 17); (4.4.6)
 

 

5.       Define a piecewise function for the following image: 

Image 

 

 

f := proc (x) options operator, arrow, function_assign; piecewise(`and`(`<=`(0, x), `<=`(x, 3)), `+`(`*`(3, `*`(x)), 1), `and`(`<`(3, x), `<=`(x, 9)), `+`(`*`(`/`(1, 6), `*`(x)), 9.5), `and`(`<`(9, x)... 

f := proc (x) options operator, arrow; piecewise(`and`(`<=`(0, x), `<=`(x, 3)), `+`(`*`(3, `*`(x)), 1), `and`(`<`(3, x), `<=`(x, 9)), `+`(`*`(`/`(1, 6), `*`(x)), 9.5), `and`(`<`(9, x), `<=`(x, 20)), `... (4.5.1)
 

 

plot(f(x), x = -5 .. 25, y = -5 .. 25, discont = true);  

Plot_2d
 

 

6.       Recreate the image, with the points, in Maple. 

 

A := [0, 1]; 1; B := [3, 10]; 1; C := [9, 11]; 1; E := [20, 20] 

 

 

 

A := [0, 1];
B := [3, 10];
C := [9, 11];
E := [20, 20]; (4.6.1)
 

with(plots); -1 

Pts := pointplot([A, B, C, E], symbolsize = 20, symbol = solidcircle);  

Plot_2d
 

crv := plot(f(x), x = -5 .. 25, y = -5 .. 25, discont = true);  

Plot_2d
 

display(crv, Pts);  

Plot_2d
 

 

7.       Determine the inverse of the function. Do you notice anything interesting about the slope values? 

 

solve(f(y) = x, y);  

piecewise(`<`(x, 1.), [], `<=`(x, 10.), [`+`(`-`(.3333333333), `*`(.3333333333, `*`(x)))], `<=`(x, 11.), [`+`(`-`(57.), `*`(6., `*`(x)))], `<=`(x, 20.), [`+`(`-`(4.444444444), `*`(1.222222222, `*`(x))... (4.7.1)
 

 

piecewise(`<`(x, 1), [], `<=`(x, 10), [`+`(`-`(`/`(1, 3)), `*`(`/`(1, 3), `*`(x)))], `<=`(x, 11), [`+`(`-`(57), `*`(6, `*`(x)))], `<=`(x, 20), [`+`(`-`(`/`(40, 9)), `*`(`/`(11, 9), `*`(x)))], `<`(20, ... (4.7.2)
 

 

 

 

finv := proc (x) options operator, arrow, function_assign; piecewise(`and`(`<`(1, x), `<`(x, 10)), `+`(`-`(`/`(1, 3)), `*`(`/`(1, 3), `*`(x))), `and`(`<`(10, x), `<=`(x, 11)), `+`(`-`(57), `*`(6, `*`(... 

finv := proc (x) options operator, arrow; piecewise(`and`(`<`(1, x), `<`(x, 10)), `+`(`-`(`/`(1, 3)), `*`(`/`(1, 3), `*`(x))), `and`(`<`(10, x), `<=`(x, 11)), `+`(`-`(57), `*`(6, `*`(x))), `and`(`<`(1... (4.7.3)
 




 

8.      Plot the function and its inverse, andy;  on the same image. 

plot([f(x), finv(x), x], x = -5 .. 20);  

Plot_2d
 

 

9.      Use symmetry to plot the function and its inverse to create the following image:Image 

 

plot([f(x), finv(x), `+`(`-`(f(x))), `+`(`-`(finv(x))), f(`+`(`-`(x))), `+`(`-`(f(`+`(`-`(x))))), finv(`+`(`-`(x))), `+`(`-`(finv(`+`(`-`(x)))))], x = -20 .. 20, color = red);  

Plot_2d