Answer
A. The domain is $(-\infty, \infty)$
B. The y-intercept is $1$
The x-intercept is $-0.74$
C. The function is not an odd function or an even function.
D. $\lim\limits_{x \to -\infty}(x+cos~x) = -\infty$
$\lim\limits_{x \to \infty}(x+cos~x) = \infty$
No asymptotes.
E. The function is increasing on the interval $(-\infty, \infty)$
F. There is no local maximum or local minimum.
G. The graph is concave down on the intervals $(\frac{3\pi}{2}+2\pi~n, \frac{5\pi}{2}+2\pi~n)$, where $n$ is an integer.
The graph is concave up on the intervals $(\frac{\pi}{2}+2\pi~n, \frac{3\pi}{2}+2\pi~n)$, where $n$ is an integer.
The points of inflection are $(\frac{\pi}{2}+\pi~n, \frac{\pi}{2}+\pi~n)$, where $n$ is an integer.
H. We can see a sketch of the curve below.
Work Step by Step
$y = x+cos~x$
A. The function is defined for all real numbers.
The domain is $(-\infty, \infty)$
B. When $x=0$, then $y = (0)+cos~0 = 1$
The y-intercept is $1$
When $y = 0$:
$x+cos~x = 0$
$cos~x = -x$
$x \approx -0.74$
The x-intercept is $-0.74$
C. The function is not an odd function or an even function.
D. $\lim\limits_{x \to -\infty}(x+cos~x) = -\infty$
$\lim\limits_{x \to \infty}(x+cos~x) = \infty$
There are no asymptotes.
E. We can find values of $x$ such that $y' = 0$:
$y' = 1-sin~x = 0$
$sin~x = 1$
$x = \frac{\pi}{2}+2\pi~n,$ where $n$ is an integer
When $0 \lt x \lt \frac{\pi}{2}$ or $\frac{\pi}{2} \lt x \lt 2\pi$, then $y' \gt 0$
The function is increasing on the interval $(-\infty, \infty)$
F. Since the function is increasing on all intervals, there is no local maximum or local minimum.
G. We can find the values of $x$ such that $y'' = 0$:
$y'' = -cos~x = 0$
$cox~x = 0$
$x = \frac{\pi}{2}+\pi~n,$ where $n$ is an integer
The graph is concave down on the intervals $(\frac{3\pi}{2}+2\pi~n, \frac{5\pi}{2}+2\pi~n)$, where $n$ is an integer.
The graph is concave up on the intervals $(\frac{\pi}{2}+2\pi~n, \frac{3\pi}{2}+2\pi~n)$, where $n$ is an integer.
When $x= \frac{\pi}{2}+\pi~n$:
$y = (\frac{\pi}{2}+\pi~n)+cos~(\frac{\pi}{2}+\pi~n)$
$y = \frac{\pi}{2}+\pi~n+0$
$y = \frac{\pi}{2}+\pi~n$
The points of inflection are $(\frac{\pi}{2}+\pi~n, \frac{\pi}{2}+\pi~n)$, where $n$ is an integer.
H. We can see a sketch of the curve below.