Calculus- Early Transcendentals, 2021a

2.8. Additional Exercises 71
Figure 2.4: Geometric definitions. Here, t is twice the shaded area in each figure.
Since coshx > 0, sinhx is increasing and hence one-to-one, so sinhx has an inverse, arcsinhx. Also,
sinhx > 0whenx > 0, so coshx is injective on [0,∞) and has a (partial) inverse, arccoshx. The other
hyperbolic functions have inverses as well, though arcsechx is only a partial inverse.
Exercises for 2.7
Exercise 2.7.1 Show that the range of sinhx is all real numbers. (Hint: show that if y = sinhx then
x = ln(y + √ y 2 + 1).)
Exercise 2.7.2 Show that the range of tanhxis(−1,1). What are the ranges of coth, sech, and csch? (Use
the fact that they are reciprocal functions.)
Exercise 2.7.3 Prove that for every x,y ∈ R, sinh(x + y)=sinhxcoshy + coshxsinhy. Obtain a similar
identity for sinh(x − y).
Exercise 2.7.4 Prove that for every x,y ∈ R, cosh(x + y) =coshxcoshy + sinhxsinhy. Obtain a similar
identity for cosh(x − y).
Exercise 2.7.5 Show that sinh(2x)=2sinhxcoshx and cosh(2x)=cosh 2 x+sinh 2 x for every x. Conclude
also that (cosh(2x) − 1)/2 = sinh 2 x.
Exercise 2.7.6 What are the domains of the six inverse hyperbolic functions?
Exercise 2.7.7 Sketch the graphs of all six inverse hyperbolic functions.
2.8 Additional Exercises
Exercise 2.8.1 If f (x)= 1
( ) 1
x − 1 , then which of the following is equal to f ?
x