Which of the following is implicit functions -

Options

A. `y = x^3 + 4x^2 + 5x`
B. `x + y =1`
C. `y = 1-x`
D. None of these

Answer

To determine which of the given options represents an implicit function, we first need to understand the definition of an implicit function. An implicit function is one where it is not possible to express one variable solely in terms of the other variable. In other words, we cannot isolate \( y \) as a function of \( x \) or \( x \) as a function of \( y \).

### Step-by-Step Solution:

1. **Identify the Functions**:
We need to analyze the given functions to check if they can be expressed explicitly.

2. **Examine Each Function**:
- **Function 1**: Let's say this function is \( f_1(x) = g_1(y) \). If we can express \( y \) in terms of \( x \) or vice versa, it is not an implicit function.
- **Function 2**: Similarly, for \( f_2(x) = g_2(y) \), we check if we can isolate \( y \) or \( x \).
- **Function 3**: For \( f_3(x) = g_3(y) \), we perform the same check.

3. **Conclusion**:
If none of the functions can be expressed in the form of \( y = f(x) \) or \( x = g(y) \), then these functions are implicit. If at least one can be expressed in such a way, it is not an implicit function.

4. **Final Answer**:
After analyzing all the functions, if we find that all of them can be expressed in terms of one variable, we conclude that none of the options are implicit functions.