Preparatory SAC Questions - FUNCTIONS

[dinosaur93]

1. If , then state the range of f.


2. Find the equation of the tangent and the perpendicular to the curve at (2, 17). How do you find the perpendicular via CAS Calculator or by hand?


3. The graphs of and intersects at 2 distinct point for Is this right?


4. Given that

Given that , find f(a + 1) in terms of 'a' for all other values of 'a'. (HINT: Express your answer as a hybrid function)


5. A function with the equation having a period of 12, a range of [-9, 15] and contains the point (2, 15). Determine the values of 'A', 'n', 'b' and 'c' given they are all positive.


6. The function is altered by the following transformations Write the equation of the transformed function.


7.

a.) Find D, a suitable domain of g(x) such that g^-1 (x) exists.

b.) State the domain and range of g for the answer in a)
dom_g = ____
ran_g = ____

c.) State the equation of g^-1 (x), using the proper function notation.

d.) Find the exact value of the intersection point.

 

8. If the function is divisible by (t - f)(f - t) and (ht + g), then identify the incorrect statement

A. a equals -h
B. d equals -f²g
C. The three t intercepts are located at t = -f, f, and -g/h
D. The cubic touches the t-axis at t = f
E. f(0) - d

Am I right?


9. The graph of is sketched. The domain and range respectively are:
A. R and R
B. R and R+
C. R and [2, \infty)
D. R and (2, \infty)
E. (2, \infty) and R


10. The function f: [\frac{-1}{2}, \frac{1}{2}] \to R, f(x) = 2 tan (3\pi x)  has an asymptotes for this restricted domain at ____?____ Show working out...


11. Define the hybrid function of the graph .   

12.  A probe use to measure the temperature of a substance. The values recorded follow the function , where t is time in minutes and T is temperature in ⁰C. State he initial temperature and the next 2 times the substance will have this temperature

Is it 70⁰ C initial temperature and 23 mins, 46 mins?

[fred42]

1.  Sketch the graph as normal, remembering to put in the endpoints. Range = [0,84).
2.  Define f(x) = (your function). Go to menu 9 4 for tangent (or 5 for normal). In the bracket after tangntLine(f(x),x=2). This will give 18x-19. Just put in y= in front. By hand, the gradient of the perp will be 1/18 and use the given point to find the equation.

3.  Equating will give a/x = x + 2
                                  a = x^2 + 2x
                                  x^2 +2x - a = 0
                                  x = (-2 +/- sqrt(4 + 4a))/2
                                  x = (-2 +/- 2sqrt(1 + a))/2
                                  x = -1 +/- sqrt(1 + a)  therefore 2 solutions for all values a > -1.

4.  f(a + 1) ={1/(a + 1 - 2)^2,  a + 1>2
                       -4,                     a + 1<=2

                  ={1/(a - 1)^2,  a>1
                      -4,                 a<1            (since a is not equal to 1)

5.   Period = 12 = 2pi/n so n = pi/6
      range means there is a vertical translation of (-9 + 15)/2 = 3 so c = 3 and amp = 12. Since A > 0, A = 12.
     x = 2 gives a y = 15 which is the max so pi/6(2 - b) = 0 so b = 2.

6.  x' = 2x so x = x'/2 and y' = -y + 2 so y = -y' + 2. Substituting gives
   -y + 2 = (x/2)^2 so y = -x^2/4 + 2

7.  a) Need either left half or right half. D = [1,inf)
     b)  Range = [2,inf)
     c)  g^-1:[1,inf) arrow [2,inf), g^-1 = sqrt(x - 2) + 1
     d) Normally you would solve (x - 1)^2 + 2 = x as both graphs meet on the y = x line IF they meet. These graphs do not meet. You've made an error.

8.  f(t) = -(t - f)^2(ht + g).
    When this is expanded, the first term will be -ht^3 so A is correct.
    The last term will be -f^2g so B is correct.
    As there is a repeated factor, there will be only 2 intercepts at t = f and -g/h so C is INCORRECT
    D is correct (see above)
    Of course E is correct.

9.  As hybrid:
    y={-(x - 1) - (x - 1) = -2x, x<-1
        {-(x - 1) + (x +1) = 2, -1<=x<1
        {x - 1 + x + 1 = 2x,x>=1
So Dom = R and ran = [2,inf) C

10.  You have this wrong and it's too late at night for me to suggest that pi/3 should be the coeff.

     Solve cos(pix/3) = 0 for asymptotes.

11.  Use method in Q9.
12.   T(0) = 36. The next time will be a period later  at t = 23 minutes and again at t = 46 minutes.