Let the function `f:R to R` be defined by `f(x)=cos x, AA x in R.` Show that `f` is neither one-one nor onto.

Explanation

To show that the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \cos x \) is neither one-one nor onto, we will analyze each property step by step.

### Step 1: Show that \( f \) is not one-one

To prove that \( f \) is not one-one, we need to find at least two different values of \( x \) such that \( f(x_1) = f(x_2) \).

1. Consider \( x_1 = \frac{\pi}{2} \) and \( x_2 = -\frac{\pi}{2} \).
2. Calculate \( f\left(\frac{\pi}{2}\right) \):
\[ f\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) = 0 \]
3. Calculate \( f\left(-\frac{\pi}{2}\right) \):
\[ f\left(-\frac{\pi}{2}\right) = \cos\left(-\frac{\pi}{2}\right) = 0 \]
4. Since \( f\left(\frac{\pi}{2}\right) = f\left(-\frac{\pi}{2}\right) \) and \( \frac{\pi}{2} \neq -\frac{\pi}{2} \), we conclude that \( f \) is not one-one.

### Step 2: Show that \( f \) is not onto

To prove that \( f \) is not onto, we need to show that there exists at least one value in the codomain \( \mathbb{R} \) that is not the image of any \( x \in \mathbb{R} \).

1. The range of the cosine function \( f(x) = \cos x \) is limited to the interval \([-1, 1]\).
2. Therefore, any real number outside this interval cannot be achieved by \( f \). For example, consider the value \( -2 \).
3. We need to check if there exists an \( x \in \mathbb{R} \) such that \( f(x) = -2 \). However, since \( -2 < -1 \), it is impossible for \( f(x) = -2 \) because the cosine function never takes values less than -1.
4. Thus, there is no \( x \in \mathbb{R} \) such that \( f(x) = -2 \), which shows that \( f \) is not onto.

### Conclusion

Since we have shown that \( f \) is not one-one and not onto, we conclude that the function \( f(x) = \cos x \) is neither one-one nor onto.

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