Sketch a graph of the following functions gx 2x2 - x -15

Question

1. Find the domain of the following functions:

f(x) = x²/x²-5x+4,            

                x³-x/√(x-4),  x < 5;

g(x) =

                100,              x ≥ 10.                            

2. Sketch a graph of the following functions:

                x²,     x ≥ 0;

f(x) =

                -x²,    x < 0

g(x) = |2x² - x -15|.

Specify whether each function is even , odd, or neither. Write down the intervals in which each function is increasing.

3. Sketch the following graphs:

f₁(x) = sinx,         f₂(x) = sin(x - π/2),           f₃(x) = sin(x - π/2) + 3

and

g₁(x) = e^x,            g₂(x) = e^x-2,                          g3(x) = e^x-2 + 3. 

4. Graph the ellipse (x-1)² + 9(y-2)² = 1. Use the vertical line test to see if the curve is the graph of a function.

5. Let f(x) = sinx and g(x) = log₂x. Write out f o g and g o f and the domains of these compositions.  

6. Solve the following equations for x:

i) log₂(x+3) - log₄(x+3) = 2;

ii) 3^x2 = 9^4x.

7. Let f(x) = ln(2+x³). Find f^-1(y). Write down the domain of f and f^-1.

8. Simplify the expression sin(cos^-1x), x ∈ (-1, 1).

9. Given that lim_x→0 sinx/x =1. Evaluate lim_x→0 sinπx/x.  

10. Given the graph of f below. Determine lim_x→-∞f(x), lim_x→0-f(x), lim_x→0+f(x), lim_x→0 f(x), lim_x→1-f(x), lim_x→1+f(x), lim_x→1f(x), lim_x→2f(x), lim_x→3-f(x), lim_x→3+f(x), lim_x→3f(x), lim_x→∞f(x).