Question

If y is expressed in terms of a variable x as $y = f(x)$, then y is called
A. Explicit function
B. Implicit function
C. Linear function
D. Identity function

Answer 1

We define each function given in the options and check which of the given functions represents a variable in terms of other variables in given way. Here one variable is stated as a function of another variable where one is independent variable and other is dependent variable.

• Independent variable: A variable that does not change with effect to any other variable is called independent variable.
• Dependent variable: A variable that changes with any change to another variable is called Dependent variable. Dependent variable is dependent on the independent variable for its value.

Complete step-by-step solution:
We define each function given in the four options separately.
A. Explicit function:
A function in which dependent variables can be written entirely in terms of the independent variable is called an Explicit function. Here dependent variable is on one side of the equation and rest other terms in terms of independent variable are on the other side of the equation. For example: $y = 2{x^2} - 2x + 7$
B. Implicit function:
A function having an equation of the form $R = 0$; where R is a function of several variables. For example: Equation of circle is implicit equation i.e. ${x^2} + {y^2} - 1 = 0$
C. Linear function:
A function is a function of a linear polynomial i.e. a polynomial having degree 0 or 1. The graph of linear function is always a straight line. For example: $y = 3x + 2$
D. Identity function:
A function is a function which when used with any other function gives the same value as the function. So, for a real-valued function ‘f’ we can write $f:R \to R$such that $f(x) = x$, then ‘f’ is an identity function. For example: $f(7) = 7$
From the four definitions above we see that the only definition matching our requirement given in the statement is of Explicit function. We want y to be expressed in terms of a variable x as$y = f(x)$ and we know the explicit function represents a variable ‘y’ explicitly in terms of another variable ‘x’.

$\therefore$Option A is correct.

Note: Students are likely to make the mistake of choosing the option Linear function as many students don’t know what are explicit and implicit functions. Also, many students think a linear function can be molded into a function where we have one variable in term of another so that is the answer, but we are looking at the initial equations, else we can mold all the equations in some or other way to transform into one variable at one side and other variable on other side.