f(x)=x² + 4sinx\nchoose the correct graph below.\na.\nall graphs are -2π, 2π by -5, 20.\ndoes the function…
f(x)=x² + 4sinx\nchoose the correct graph below.\na.\nall graphs are -2π, 2π by -5, 20.\ndoes the function appear to be periodic?\nyes\nno
Answer
Explanation:
Step1: Analyze the components of the function
The function $f(x)=x^{2}+4\sin x$ is a sum of a quadratic function $y = x^{2}$ and a sine - function $y = 4\sin x$. The quadratic function $y=x^{2}$ is a parabola opening upwards with the vertex at the origin $(0,0)$. The sine function $y = 4\sin x$ has an amplitude of 4 and a period of $2\pi$. When we add them together, the quadratic term $x^{2}$ will dominate as $|x|$ gets large.
Step2: Recall the property of periodic functions
A function $y = f(x)$ is periodic if there exists a positive real number $T$ such that $f(x + T)=f(x)$ for all $x$ in the domain of $f$. For the function $f(x)=x^{2}+4\sin x$, consider $f(x + T)=(x + T)^{2}+4\sin(x + T)=x^{2}+2Tx+T^{2}+4\sin(x + T)$. If $T$ is a non - zero positive number, $x^{2}+2Tx+T^{2}+4\sin(x + T)\neq x^{2}+4\sin x$ for all $x$. The non - periodic term $x^{2}$ makes the whole function non - periodic.
Answer:
No