Assignment:
Inverse function problems
Q1. For the following problems a function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one to one.
1) X = 1, 2, 3, 4, 5, 6
F(x) = 1.5, 2.0, 3.6, 5.3, 2.8, 2.0
2) f(x) = ½(x+5)
3) g(x) = SQRT(x)
4) f(x) = 1+4x-x^2
5) h(x) = x^4+5
6) h(x) = x^4+5, 0 is <= x is <= 2
7) Please define a function and what is, what is its purpose in math? I know you say have f(2), well, you plug in 2 for x and solve and y is or ='s f(x), but what is the difference between x and f(x)?
Q2. If f is a one to one function such that f(2) = 9, what is f^-1(9)?
1) If f(x) = x+cosx, find f^-1(1)
2) if h(x) = x + SQRT(x) find h^-1(6)
3) The formula C=(5/9)(F-32), where F is >= -459.67, expresses the Celsius temp. C as a function of the Fahrenheit temp F. Find a formula of the inverse function and interpret it. What is the domain of the inverse function?
4) Find a formula for the inverse functions.
a) f(x)=3-2x
b) f(x) = SQRT(10-3x)
c) y= (1-SQRT(x))/ (1+SQRT(x))
d) y = 2x^3 + 3
Q3. For the next 3 problems show that
a) f is one to one.
b) Use theorem 7 to find g'(a), where g = f^-1.
c) Calculate g(x) and state the domain and range of g.
d) Calculate g'(a) from the formula in part c and check that it agrees with the result of part b.
e) Sketch the graphs of f and g on the same axes.
1) f(x) = x^3, a=8
2) f(x) = SQRT(x-2), a=2
3) f(x) =9-x^2, 0<= x<=3, a=8
Q4. Find (f^-1)'(a)
F(x) =x^3+x+1
Q5. Find (f^-1)'(a)
F(x) =3+ x^2 + tan(pi(x)/2), -1
Q6. Suppose g is the inverse function of f and f(4) = 5, f'(4)=(2/3), find g'(5).
Provide complete and step by step solution for the question and show calculations and use formulas.