Question
Which pairs of function is identical ?
A.
$$f\left( x \right) = \sqrt {{x^2}} ,g\left( x \right) = x$$
B.
$$f\left( x \right) = {\sin ^2}x + {\cos ^2}x,g\left( x \right) = 1$$
C.
$$f\left( x \right) = \frac{x}{x},g\left( x \right) = 1$$
D.
None of these
Answer :
$$f\left( x \right) = {\sin ^2}x + {\cos ^2}x,g\left( x \right) = 1$$
Solution :
For checking equal function
$$\left( A \right)$$ Domain of $$f\left( x \right) = R$$ but range $$ = \left[ {0,\infty } \right)$$
Domain of $$g\left( x \right) = R,$$ range $$ = R$$
Domain same but range is different so it is not an equal function.
$$\left( B \right)$$ Domain of $$f\left( x \right) = R$$
Domain of $$g\left( x \right) = R$$
Domain and range both same so it is an equal function.
$$\left( C \right)$$ Domain of $$f\left( x \right) = R - \left\{ 0 \right\}$$
Domain of $$g\left( x \right) = R$$
Not equal function as domain is different.