Question

Which of the following function is implicit function ? xy – cos(x + y) = 0 y = x3 y = log(x2 + x + 1) y = |x|

• A. xy – cos(x + y) = 0
• B. y = x3
• C. y = log(x2 + x + 1)
• D. y = |x|

Answer 1

Answer: (A)

xy – cos(x + y) = 0

Answer 2

An implicit function is a function where the dependent variable (y) is not explicitly expressed solely in terms of the independent variable (x). Instead, the relationship between x and y is given by an equation of the form $F(x, y) = 0$.

Let's analyze each given option:

1. $xy – \cos(x + y) = 0$ In this equation, y is not isolated on one side. It appears in the product with x ($xy$) and within the cosine function ($\cos(x+y)$). It is not straightforward to express y explicitly as a function of x. This fits the definition of an implicit function.

2. $y = x^3$ Here, y is explicitly defined in terms of x. This is an explicit function.

3. $y = \log(x^2 + x + 1)$ Here, y is explicitly defined in terms of x. This is an explicit function.

4. $y = |x|$ Here, y is explicitly defined in terms of x. This is an explicit function.

Based on the definitions, only the first option represents an implicit function.