Question

Question was provided via image upload.

Explanation:

Identify the mathematical expressions

The image lists four mathematical functions of the variable \(t\):

1. \(f_1(t) = \cos^2(\omega t)\)
2. \(f_2(t) = 3 + t^2 + \sin(\pi t)\)
3. \(f_3(t) = a + bt + ct^2\)
4. \(f_4(t) = A\cos(\omega t) + B\sin(\omega t)\) (partially cut off as \(B\sin\))

We will analyze the symmetry properties (even, odd, or neither) of each function.

Analyze the first function

For \(f_1(t) = \cos^2(\omega t)\):
Substitute \(-t\) for \(t\):

$$ f_1(-t) = \cos^2(\omega (-t)) = (\cos(-\omega t))^2 $$

Since cosine is an even function, \(\cos(-\theta) = \cos(\theta)\):

$$ f_1(-t) = (\cos(\omega t))^2 = \cos^2(\omega t) = f_1(t) $$

Thus, \(f_1(t)\) is an even function.

Analyze the second function

For \(f_2(t) = 3 + t^2 + \sin(\pi t)\):
Substitute \(-t\) for \(t\):

$$ f_2(-t) = 3 + (-t)^2 + \sin(\pi (-t)) = 3 + t^2 - \sin(\pi t) $$

Since \(f_2(-t)
eq f_2(t)\) and \(f_2(-t)
eq -f_2(t)\), this function is neither even nor odd.

Analyze the third function

For \(f_3(t) = a + bt + ct^2\):
Substitute \(-t\) for \(t\):

$$ f_3(-t) = a + b(-t) + c(-t)^2 = a - bt + ct^2 $$

• If \(b = 0\), then \(f_3(-t) = a + ct^2 = f_3(t)\), which is even.
• If \(a = 0\) and \(c = 0\), then \(f_3(-t) = -bt = -f_3(t)\), which is odd.
• In general, if \(b

eq 0\) and at least one of \(a\) or \(c\) is non-zero, the function is neither even nor odd.

Analyze the fourth function

For \(f_4(t) = A\cos(\omega t) + B\sin(\omega t)\):
Substitute \(-t\) for \(t\):

$$ f_4(-t) = A\cos(-\omega t) + B\sin(-\omega t) = A\cos(\omega t) - B\sin(\omega t) $$

• If \(B = 0\), the function is even.
• If \(A = 0\), the function is odd.
• If both \(A

eq 0\) and \(B
eq 0\), the function is neither even nor odd.

Answer:

The symmetry analysis of the functions shown in the image is as follows:

1. \(f_1(t) = \cos^2(\omega t)\)

• Symmetry: Even function, because \(f_1(-t) = f_1(t)\).

1. \(f_2(t) = 3 + t^2 + \sin(\pi t)\)

• Symmetry: Neither even nor odd, because \(f_2(-t)

eq f_2(t)\) and \(f_2(-t)
eq -f_2(t)\).

1. \(f_3(t) = a + bt + ct^2\)

• Symmetry: Generally neither even nor odd. It is even if \(b = 0\), and odd if \(a = c = 0\).

1. \(f_4(t) = A\cos(\omega t) + B\sin(\omega t)\)

• Symmetry: Generally neither even nor odd. It is even if \(B = 0\), and odd if \(A = 0\).