Define `f: R to R` by
`f(x) = (sin^(2)x + cos^(4)x)/(cos^(2)x + sin^(4)x)`, then the range of f consists of exactly ………….. Element(s).

Answer

To find the range of the function \( f: \mathbb{R} \to \mathbb{R} \) defined by

\[ f(x) = \frac{\sin^2 x + \cos^4 x}{\cos^2 x + \sin^4 x}, \]

we will simplify the expression step by step.

### Step 1: Rewrite the function
We start with the given function:

\[ f(x) = \frac{\sin^2 x + \cos^4 x}{\cos^2 x + \sin^4 x}. \]

### Step 2: Use trigonometric identities
Recall the identity \( \sin^2 x + \cos^2 x = 1 \). We can express \( \sin^2 x \) in terms of \( \cos^2 x \):

\[ \sin^2 x = 1 - \cos^2 x. \]

### Step 3: Substitute in the function
Now substitute \( \sin^2 x \) in the function:

\[ f(x) = \frac{(1 - \cos^2 x) + \cos^4 x}{\cos^2 x + (1 - \cos^2 x)^2}. \]

### Step 4: Simplify the numerator
The numerator becomes:

\[ 1 - \cos^2 x + \cos^4 x = 1 - \cos^2 x + \cos^4 x. \]

### Step 5: Simplify the denominator
The denominator becomes:

\[ \cos^2 x + (1 - 2\cos^2 x + \cos^4 x) = 1 - \cos^2 x + \cos^4 x. \]

### Step 6: Factor out common terms
Notice that both the numerator and denominator are the same:

\[ f(x) = \frac{1 - \cos^2 x + \cos^4 x}{1 - \cos^2 x + \cos^4 x} = 1. \]

### Conclusion
Since \( f(x) \) simplifies to 1 for all \( x \), the function is constant. Therefore, the range of \( f \) consists of exactly one element, which is:

\[ \text{Range of } f = \{1\}. \]

### Final Answer
The range of \( f \) consists of exactly **1 element**.

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