Math 1P97. Assignment 1 -Find the domains and ranges of the following functions
Math 1P97. Assignment 1 (4%).Student’s name:Total mark:NOTE: Students are expected to complete ALL questions on the assignment. However, only a subset ofquestions will be considered for marking. Marks will be deducted for incomplete assignments. Show allyour work.1. Find the domains and ranges of the following functions:a) f (x) =| 1 − x2 | +1b) f (x) =√9 − x2c) f (x) = 24−x2<4 marks> Find the rules for the composite functions (g ◦ f )(x) and (f ◦ g)(x)2.a)If f (x) = 5x and g(x) = cos 2x,b)If f (x) = 4 − 4x + x2 and g(x) =c)If f (x) =√x2,x−4×2 + 1 and g(x) = ln x,3. An office building worth $16, 172, 800 million when completed in 1990 was depreciated linearly over50 years to $10, 000, 000.a) Express the book value of the building V (n) as a function of year starting from 1990: n =0, V (0) = 16, 172, 800 and sketch a graph.b) What were the book values of the building in 2010 and 2015?c) Find the rate at which the building is being depreciated each year.4. Find the indicated limits, if they exist.a)ex (x − 2)=x→1 x2 − x − 6lim2x2 − 7x + 3b) lim 2=x→3 x − x − 6c)d)e)1 − cos 5x=x→0 1 − cos 3xlimlim x2 + 1 x2x→∞x2=ln(1 − 3x)=x→0xlim5. Use the graph of the given function f to determine lim f (x) or lim f (x) and lim f (x) at thex→ax→a+x→a−following values of a:a) a = −∞ :lim f (x) =x→−∞b) a = −8c) a = −4d) a = −2e) a = 2f) a = 6g) a = ∞.6. Let f =4 − x2 ,x3 − 2×2 + a,if x ≤ 2Find the value of a that will make f continuous on (−∞, ∞).if x > 27. (Commissions, problem 68) The base monthly salary of a salesman working on commission is $22, 000.For each $50, 000 of sales beyond $100, 000, he is paid a $1000 commission.a) Sketch a graph y = f (x) showing his earnings y as a function of the level of his sales x (inthousands).b) Determine the values of x for which the function f is discontinuous and find their one-sidedlimits.8. (Energy expended by a fish, problem 72) Suppose a fish swimming a distance of L ft at a speed ofυ ft/sec relative to the water and against a current flowing at the rate of u ft/sec (u < υ) expendsa total energy given byaLυ 3,E(υ) =υ−uwhere E is measured in foot-pounds (ft-lb) and a is a constant.a)Evaluate lim E(υ) and interpret your result.b)Evaluate lim E(υ) and interpret your result.υ→u+υ→∞
