If the relation `f(x)={(2x-3",",x le 2),(x^(3)-a",",x ge2):}` is a function, then find the value of a.

Explanation

To determine the value of \( a \) such that the relation \( f(x) = \{(2x - 3, x \leq 2), (x^3 - a, x \geq 2)\} \) is a function, we need to ensure that the outputs from both parts of the function match at the point where they transition, which is at \( x = 2 \).

### Step-by-Step Solution:

1. **Identify the two parts of the function**:
- For \( x \leq 2 \): \( f(x) = 2x - 3 \)
- For \( x \geq 2 \): \( f(x) = x^3 - a \)

2. **Evaluate \( f(x) \) at \( x = 2 \)**:
- From the first part (when \( x \leq 2 \)):
\[ f(2) = 2(2) - 3 = 4 - 3 = 1 \]
- From the second part (when \( x \geq 2 \)):
\[ f(2) = 2^3 - a = 8 - a \]

3. **Set the two expressions equal to each other**:
Since \( f(2) \) must be the same from both parts for the relation to be a function, we set them equal:
\[ 1 = 8 - a \]

4. **Solve for \( a \)**:
Rearranging the equation gives:
\[ a = 8 - 1 \]
\[ a = 7 \]

Thus, the value of \( a \) is \( 7 \).

### Final Answer:
The value of \( a \) is \( 7 \).

---