topic
Analyzing the function \ (f (x) = ln (x + \ sqrt {1 + x ^ {2}}) \) parity.
Resolve
The title used knowledge
\(log_{a}(MN)=log_{a}M+log_{a}N\)
In the MATLAB ( The following code in MATLAB 9.1.0.441655 (R2016b input code in the following) by Test):
x=0:0.01:10;
semilogy(x,log(x))Can be plotted (y = ln (x) \ ) \ images:
figure 1
The image can be seen, the natural logarithm \ (ln (x) \) only \ ((0, + \ infty ) \) range, there are defined, does not meet the logarithmic function or functions for an even number of "domain \ ( X \) requirements on the origin of symmetry ". However, the function title can be seen as a compliance function, therefore, we need to combine \ (g (x) = x + \ sqrt {1 + x ^ {2}} \) of the domain to determine \ (f (x ) \) in the domain.
because:
\(\sqrt{1+x^{2}}>\sqrt{x^{2}}>|x|>0\)
then:
When \ (x \ in (- \ infty, + \ infty) \) when \ (x + \ sqrt {1 + x ^ {2}}> 0 \) satisfies the natural logarithm function \ (ln (x) \) of domain requirements, but, when the \ (x = 0 \) when \ (f (x) = LN (. 1) = 0 \) time, satisfies the odd function "if f (x) is defined at the origin, requirements f (0) = 0 ".
Here, the problem solving domain, the following functions are to be solved is about \ (Y \) axially symmetric or symmetric about the origin of the problem.
due to:
\(f(x)=ln(x+\sqrt{1+x^{2}})\)
\(f(-x)=ln(-x+\sqrt{1+x^{2}})\)
then:
\(f(x)+f(-x)=ln(\sqrt{1+x^{2}}+x)+ln(\sqrt{1+x^{2}}-x)=ln[(\sqrt{1+x^{2}}+x)(\sqrt{1+x^{2}}-x)]=ln(1+x^{2}-x^{2})=ln(1)=0\)
Defined above calculation results for odd function, therefore, \ (F (X) = LN (X + \. 1 + sqrt {X} ^ {2}) \) is an odd function.
Furthermore, the use WolframAlpha draw function \ (f (x) = ln (x + \ sqrt {1 + x ^ {2}}) \) image is as follows:
Figure 2. Images from https://www.wolframalpha.com/
The image we can see that this is an odd function.
EOF