Question

1. If fand g are both even functions, is the product fg even? If f and g both odd functions, is fg odd? What if f is even and g is odd? Justify your answers. (10 points) Find the domain g(x) =-. (10 points) 2. of the composited function fog, where f(x)=x+ and x +1 x+2 3. Let ifx <1 g(x) = x-3 ifx >2 Evaluate each of the following, if it exists. (10 points) lim g(x) lim gx)(i) lim g(r) (iv) lim g(x)(vlimg(x) 4. Evaluate the limit lim cos, ifit exists. (10 points) 5. Use the Intermediate Value Theorem to show that there is a root of the given equationx-3-0in the specified interval ofa2). (10 points) 6. If xytee, find the value of y'at the point when x 0 (10 points) 7. Find f(x)andf"(x),wheref(x)=(x' + 2xe(5 points) 8. Evaluate the integrale')ds (Spoints) 1/2

Answer 1

f(-x)-f(x).g(-x)-g(x) then fg(-x)-f(-x)g(--(x)g(x)-fg(x) fg is even function f (-x)-x).g(-x)--g(x) then fg (-x)-f(-x)2(-x)-(-f(x))(-g(x)) =fg(x) jg is even function jg is odd function domain of f (x)-, 0 domain of g (x) =, x #-2 fog(x)-(g(x) is defined for, g(x) is defiend and fis defined at g(x) domain of fog is , x #-2 andg(x) #0 domain of fog is, R-(-1,-2) f(g(x)--f(+2)=x+2 + x+1 lim g(x) = lim x = 1 g(x) = 3 leftlimitis not eqaul to right limit so limit of g is not exist at 1 lim g(x) = lim 2-x" =-2 lim g(x) = lim x-3= 2-3=-1 left limitis not eqaul to right limit so limit of g is not exist at x=2 imx COS- 11m x' cos|--