12.1.
Otherwise, the trig functions would not be monotonic. It implies that trig functions may take the same values for different arguments. In turn, it means that the inverse functions may have more than one value for one argument. It contradicts to the definition of a function.

12.2.
a). Here one value on horizontal axis may correspond to more than one value on vertical axis.

12.3.

12.4.
On the segment [0, 1], this function is inverse to itself.

12.5.
b). The inverse function is the image of the original function over the line y = x.

12.6.
x 0. It is the interval (– , 0].

12.7
arcos x has the range [0, 180°]. Therefore,
1) arcos 0.93 = 21°
2) arcos 0.24 = 76°
3) arcos (–0.72) = 136°
4) Since 194° = 360° – 166°, and 166° < 180°, arcos (–0.97) = 166°.

12.8.
1) 4
2) 56
3) 0
4) 2
5)
6) does not exist.

12.9.
x = ± 0.45 + 2n.

12.10.
The domain is the segment [–1, 1]. The range is the segment [0, ].

12.11.
1) No, because we lost negative values of the argument.
2) No, because on this interval, the cosine is not monotonic.
3) Yes. On this interval, the cosine is monotonic and takes all values from –1 to 1.

12.12.
Take the graph of arcos x (see Fig. 3) and shift it five units to the left.

12.13.