which of the following is true for f(x) = 5cos(x) +1?\nthe period of the function is 10π.\nthe function has…

Explanation:

Step1: Recall cosine - function properties

The general form of a cosine function is $y = A\cos(Bx - C)+D$. For the function $f(x)=5\cos(x)+1$, we have $A = 5$, $B = 1$, $C = 0$, $D = 1$.

Step2: Calculate the period

The period of a cosine function $y = A\cos(Bx - C)+D$ is given by $T=\frac{2\pi}{|B|}$. Since $B = 1$, $T = 2\pi$, so the statement "The period of the function is $10\pi$" is false.

Step3: Calculate the amplitude

The amplitude of a cosine function $y = A\cos(Bx - C)+D$ is $|A|$. Here, $|A| = 5$, so the statement "The function has an amplitude of 2.5" is false.

Step4: Find the zeros

Set $f(x)=0$, so $5\cos(x)+1 = 0$. Then $\cos(x)=-\frac{1}{5}$. When $x=\frac{\pi}{2}$, $\cos(\frac{\pi}{2}) = 0$, and $f(\frac{\pi}{2})=5\times0 + 1=1\neq0$. So the statement "A zero of the function is $(\frac{\pi}{2},0)$" is false.

Step5: Determine the range

The range of the cosine function $y=\cos(x)$ is $[- 1,1]$. For the function $y = 5\cos(x)+1$, when $\cos(x)=-1$, $y=5\times(-1)+1=-4$; when $\cos(x)=1$, $y=5\times1 + 1=6$. So the range of $y = 5\cos(x)+1$ is $-4\leq y\leq6$.

Answer:

The range of the function is the set of real numbers $-4\leq y\leq6$.