Identify the type of the function `f:R to R,`
`f(x)=e^(x^(2))+cosx.`

Explanation

To identify the type of the function \( f: \mathbb{R} \to \mathbb{R} \) given by \( f(x) = e^{x^2} + \cos x \), we will analyze the function step by step.

### Step 1: Analyze the components of the function
The function consists of two parts:
1. \( e^{x^2} \)
2. \( \cos x \)

### Step 2: Determine the range of \( e^{x^2} \)
The term \( e^{x^2} \) is always positive since the exponential function is always greater than zero for any real number. Specifically:
- As \( x \) approaches \( \pm \infty \), \( e^{x^2} \) approaches \( \infty \).
- At \( x = 0 \), \( e^{0^2} = e^0 = 1 \).
Thus, the range of \( e^{x^2} \) is \( [1, \infty) \).

### Step 3: Determine the range of \( \cos x \)
The cosine function oscillates between -1 and 1 for all real numbers \( x \):
- The range of \( \cos x \) is \( [-1, 1] \).

### Step 4: Combine the ranges
Now, we combine the two ranges:
- The minimum value of \( f(x) \) occurs when \( e^{x^2} \) is at its minimum (which is 1) and \( \cos x \) is at its minimum (which is -1):
\[ \text{Minimum of } f(x) = 1 + (-1) = 0 \]
- The maximum value of \( f(x) \) occurs when \( e^{x^2} \) is at its minimum (1) and \( \cos x \) is at its maximum (1):
\[ \text{Maximum of } f(x) = 1 + 1 = 2 \]
- As \( x \) increases or decreases, \( e^{x^2} \) grows without bound, thus \( f(x) \) can take values greater than 2.

### Step 5: Determine the overall range of \( f(x) \)
From the analysis, we see that:
- The function \( f(x) \) does not take any negative values.
- The minimum value is 0, and it can take any value greater than or equal to 0.

Thus, the overall range of \( f(x) \) is \( [0, \infty) \).

### Step 6: Identify the type of function
To classify the function:
- Since \( f(x) \) can take values from 0 to \( \infty \) and does not take any negative values, it is an **into function** (not onto) because it does not cover all real numbers.

### Conclusion
The function \( f(x) = e^{x^2} + \cos x \) is an **into function**.

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