If $f(x)=\frac{2^{2 x}}{2^{2 x}+2}, x \in R$, then $f\left(\frac{1}{2023}\right)+f\left(\frac{2}{2023}\right)+\ldots+f\left(\frac{2022}{2023}\right)$ is equal to
- 1011
- 2010
- 1010
- 2011
The Correct Option is A
Approach Solution - 1
Now
Now sum of terms equidistant from beginning and end is
Sum times
Approach Solution -2
Step 1: Given function
The given function is:
\[ f(x) = \frac{2^{2x^2}}{2x+2} = \frac{4x}{4x+2}. \]
Step 2: Adding \( f(x) \) and \( f(1 - x) \)
Observe that:
\[ f(x) + f(1 - x) = \frac{4x}{4x + 2} + \frac{4(1 - x)}{4(1 - x) + 2}. \]
Step 3: Simplifying the second term We simplify the second term:
\[ f(x) + f(1 - x) = \frac{4x}{4x + 2} + \frac{4(1 - x)}{4(1 - x) + 2}. \]
Step 4: Further simplification: Now, we simplify further:
\[ f(x) + f(1 - x) = \frac{4x}{4x + 2} + \frac{4}{4x + 2} + 2(4x). \]
Step 5: Final simplification :Finally, simplifying the sum gives us:
\[ f(x) + f(1 - x) = \frac{4x}{4x + 2} + \frac{2}{2 + 4x}. \]
We see that the final expression is:
\[ f(x) + f(1 - x) = 1. \]
Step 6: Summation
Consider the summation:
\[ f(1) + f(2) + \ldots + f(2023). \] Using the property \( f(x) + f(1 - x) = 1 \), we observe that pairs of terms add up to 1. The total number of terms is 2022, so there are \( \frac{2022}{2} = 1011 \) pairs.
Step 7: Final sum
The sum of all the pairs is:
\[ \text{Sum} = 1 + 1 + \ldots + 1 \quad (1011 \text{ terms}) = 1011. \]
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Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
