  Question

# Match the conditions/expressions in Column I with statement in Column II. Let f1:R→R,f2:[0,∞]→R,f3:R→R and f4:R→[0,∞) be defined by f1(x)={|x|,if x<0ex,if x≥0f2(x)=x2;f3(x)={sinx,if x<0x, if x≥0 and f4(x)={f2[f1(x)], if x<0f2[f1(x)]−1, if x≥0 Column IColumn IIa.f4 isp.onto but not one-oneb.f3 isq.neither continuous nor one-onec.f2off1 isr.differentiable but not one-oned.f2 iss.continuous and one-one   A-r    B-p    C-s    D-q A-p    B-r    C-s    D-q A-r    B-p    C-q    D-s A-p    B-r    C-q    D-s

Solution

## The correct option is D A-p    B-r    C-q    D-s f1(x)={−x, x<0ex, x≥0 f2(x)=x2, x≥0 f3(x)={sin x, x<0x, x≥0 f4(x)={f2(f1(x)), x<0f2(f(x))−1, x≥0 Now, f2(f1(x))={x2, x<0e2x, x≥0 ⇒f4={x2, x<0e2x−1, x≥0 As f4(x) is continuous, f′4(x)={2x, x<02e2x, x>0 f;4(0) is not defined. Its range is [0,∞). Thus, range = coadmin = [0,∞), thus f4 is onto. Also, horizontal line(drawn parallel to X-axis) meets the curve more than once, thus function is not one-one.  Suggest corrections   