Find the range of the following functions: (where {.} and [.] represent fractional part and greatest integer part functions respectively)
`f(x)=x^(4)-2x^(2)+5`

Explanation

To find the range of the function \( f(x) = x^4 - 2x^2 + 5 \), we can follow these steps:

### Step 1: Rewrite the function
We start by rewriting the function in a more manageable form. Notice that we can express \( f(x) \) as:
\[ f(x) = x^4 - 2x^2 + 5 \]
We can rearrange the first two terms:
\[ f(x) = (x^2)^2 - 2(x^2) + 5 \]
Let \( y = x^2 \). Then, we can rewrite the function as:
\[ f(y) = y^2 - 2y + 5 \]

### Step 2: Complete the square
Next, we complete the square for the quadratic expression:
\[ f(y) = (y^2 - 2y + 1) + 4 = (y - 1)^2 + 4 \]
This shows that \( f(y) \) is a perfect square plus 4.

### Step 3: Analyze the minimum value
Since \( (y - 1)^2 \) is always non-negative (i.e., \( (y - 1)^2 \geq 0 \)), the minimum value of \( f(y) \) occurs when \( (y - 1)^2 = 0 \). This happens when \( y = 1 \):
\[ f(1) = 0 + 4 = 4 \]

### Step 4: Determine the maximum value
As \( y \) (or \( x^2 \)) approaches infinity, \( (y - 1)^2 \) also approaches infinity. Therefore, \( f(y) \) approaches infinity:
\[ f(y) \to \infty \text{ as } y \to \infty \]

### Step 5: Conclusion about the range
Thus, the range of the function \( f(x) = x^4 - 2x^2 + 5 \) is:
\[ [4, \infty) \]

### Final Answer
The range of the function \( f(x) = x^4 - 2x^2 + 5 \) is \( [4, \infty) \).

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