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Which of the following is true for f(x) = 5cos(x) +1?


A.) The period of the function is 10 pi

B.) The function has an amplitude of 2.5.

C.) A zero of the function is ( pi/2, 0 )

The range of the function is the set of real numbers -4≤ y ≤ 6

Verified Answer :

TaeKwonDoIsDead

The function \(f(x)=5cos(x)+1\) is the cosine curve with 5 as the amplitude and is shifted 1 units up.

Let's check all the 4 choices:

A.

The period of a function in the form \(f(x)=Acos(x)+B\) (which is the function in this problem) has period \(2\pi\). Period would have changed if there was any coefficient of x in the argument. But it doesn't so period is NOT \(10\pi\).

B.

The amplitude of a function in the form \(f(x)=Acos(x)+B\) (which is the function in this problem) is A. But A is 5 here so the amplitude is definitely NOT 2.5.

C.

A zero of the function is given as \((\frac{\pi}{2},0)\), which means that if we plug in \(\frac{\pi}{2}\) into x, it should give us 0. Let's check:

\(5cos(\frac{\pi}{2})+1\\=5(0)+1\\=1\). It's not true.

D.

We know that this function has an amplitude of 5, so the part \(5cos(x)\) ranges from -5 to 5. But this part is shifted 1 units up, so the max amplitude is \(5+1=6\) and the min is \(-5+1=-4\). So we can say that the range of the function is the set of real numbers from -4 to 6. This choice is correct.

ANSWER: Last choice, choice D.

adioabiola

The statement which is true for f(x) = 5cos(x) +1 is; Choice D; The range of the function is the set of real numbers -4≤ y ≤ 6

Range of Functions

From comparison with equation of the form;

  • f(x) = Acos(x) + B

where;A = amplitude

We can conclude that this function has an amplitude of 5, so the portion 5cos(x) of the function ranges from -5 to 5.

However, this part is translated 1 units up as denoted by the +1.

Ultimately,

  • The max amplitude is at point; -5+1 = -4 and
  • The min amplitude is at point; 5+1 = 6.

Therefore, we can conclude that the range of the function is the set of real numbers from -4 to 6.

Read more on Range of a function;

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