IM3 - Final Quiz!

Mx. Haynes @Trinity

Problem Numbers

#106 - #135

Consider f(x) = (2x - 3) / (x + 1).

a) Find the domain and range of f.

b) What are the intercepts of f?

c) Are there any holes? Are there any vertical asymptotes?

a) Domain: (-∞, -1) U (-1, ∞)

x = (2y - 3) / (y+1)

xy + x = 2y-3

xy - 2y = -x-3

(x-2)y = -x-3

y = (-x-3) / (x-2)

Range: (-∞, 2) U (2, ∞)

b) X-int: (3/2, 0)

Y-int: (0, -3)

c) No holes (nothing cancels out). Vertical asymptote at x = -1 because of the denominator.

“Does the graph have any symmetry?”

What do you even do?

Find out whether it’s an even or odd function. Recall:

Even functions are when f(x) = f(-x)

Odd functions are when f(-x) = -f(x)

lim_x→1((x³-1) / (x-1))

Find by dividing → (x-1)(x²+x+1) / (x-1)

Hole @ (1,3)

3

If degree_num > degree_denom…

If degree_num = degree_denom…

If degree_num < degree_denom…

(These are all highest degree btw)

Most likely slant asymptote. Divide to find out.

If the coefficient of numerator is a and coefficient of denominator is b, end behavior @ y = a/b

End behavior @ y = 0

Why end behavior? Why not horizontal asymptote.

Graph can still pass through

f(x) = (x³-3x+2) / (x²-1)(x-3)

a) Factor and reduce the fraction. Find the coordinates of any holes.

b) Find the x- and y-intercepts of the function.

c) Find the equation of all vertical asymptotes, if any.

d) Find the equations of horizontal or slant asymptotes, if any.

a) ((x-1)(x²+x-2)) / ((x-1)(x+1)(x-3)); hole at (1,0)

b) X-ints: (1,0), (-2,0)

Y-int: (0, 2/3)

c) Vert. asymps @ x = 3 and x = -1

d) d_num = d_denom, coeffs = 1/1; end behavior @ y = 1

Find a function for a graph with x-intercepts (-2,0), (0,0), (1,0), (3,0) and (5,0) and vertical asymptotes at x = -5/2 and x = 11/2.

x(x+2)(x-1)(x-3)(x-5) / (2x+5)(2x-11)

Types of discontinuities

No discontinuity (line, parabola)

Removable discontinuity (hole)

Infinite discontinuity (asymptote)

Consider the function c(x) = (x+1) / (x-3).

a) Is the function continuous? If not, what kind of discontinuity does it have.

b) Find lim_x→3⁺ (x+1)/(x-3).

c) Find lim_x→3^- (x+1)/(x-3).

d) Find lim_x→3 (x+1)/(x-3).

a) No, infinite discontinuity (vertical asymptote at x = 3).

b) (4+1)/(4-3) = 5

(3.5+1)/(3.5-3) = 9/2 / 1/2 = 9

∞

c) (2+1)/(2-3) = -3

(2.5+1)/(2.5-3)= 7/2 / -1/2 = -7

-∞

d) DNE [two values]

Finding limits of functions

1) See if there is a hole; if so, cross out the removable discontinuity and plug the number back into what’s left to see what its y-value is.

2) If you can’t factor and find a hole, plug in values that approach the value you’re supposed to approach. For example, if you’re looking for the limit as x → 0, test -1 and -1/2 to see how it changes as you approach. If it gets bigger, assume ∞. If it gets smaller, assume -∞.

Practice? Redo 134.