#1
For the following piecewise-defined function, find (a) f(-5) (b) f(-1), (c) f(0) and (d) f(3)
f(x)=(x-2, if x< 3)
(5-x, if x≥ 3)
(a) f(-5) = -5 – 2 = -7
(b) f(-1) = -1 – 2 = -3
(c) f(0) = 0 – 2 = -2
(d) f(3) = 5 – 3 = 2
#2
Without graphing, determine whether each equation has a graph that is symmetric with
respect to the x-axis, the y-axis, the origin, or none of these.
(a) Y^2=-6/x^2
Symmetric with respect to the y-axis
(b) Y=x+15
None of these
#3
Graph the function y=|x+3|+2
The graph looks like this:
#4
Let f(x)=x^2+3 and g(x)=-2x+6. Find each of the following.
(a)(f+g)(-5)
(f + g)(x) = x2 + 3 – 2x + 6 = x2 – 2x + 9
(f + g)(-5) = (-5)2 – 2(-5) + 9 = 25 + 10 + 9 = 44
(b)(fg)(-3)
(fg)(x) = (x2 + 3)(-2x + 6) = -2x3 + 6x2 – 6x + 18
(fg)(-3) = -2(-3)3 + 6(-3)2 – 6(-3) + 18
(fg)(-3) = -2(-27) + 6(9) + 18 + 18
(fg)(-3) = 54 + 54 + 18 + 18
(fg)(-3) = 144
(c) (f/g)(5)
(f/g)(x) = (x2 + 3)/(-2x+6)
(f/g)(5) = (52 + 3)/(-2(5) + 6) = (25 + 3)/(-10 + 6)
(f/g)(5) = 28/-4
(f/g)(5) = -7
#5
For the function defined as f(x)=4x+11, Find (a) f(x+h), (b) f(x+h)-f(x) and (c) f(x+h)f(x)/h
(a) f(x + h) = 4(x + h) + 11 = 4x + 4h + 11
(b) f(x + h) – f(x) = (4x + 4h + 11) – (4x + 11) = 4x – 4x + 4h + 11 – 11 = 4h
(c) [f(x + h) – f(x)]/h = 4h/h = 4
#6
Given functions of f and g, find (a) (f o g)(x) and its domain and (b) (g o f)(x) and its
domain.
f(x)=x+2, g(x)=x^4+x^2-3x-4
(a) (f o g)(x) = (x4 + x2 – 3x – 4) + 2 = x4 + x2 – 3x – 2
The domain is “x is a real number”
(b) (g o f)(x) = (x + 2)4 + (x + 2)2 – 3(x + 2) – 4
(g o f)(x) = (x4 + 8x3 + 24x2 + 32x + 16) + (x2 + 4x + 4) – 3x – 6 – 4
(g o f)(x) = x4 + 8x3 + 25x2 + 33x + 10
The domain is “x is a real number”
#7
Express f(x) in the form f(x)=(x-k)q(x)+r for the given value of k.
F(x)=-x^3+x^2+3x-2; k=2
Dividing –x3 + x2 + 3x – 2 by x – 2, using polynomial long division, gives
a result of –x2 – x + 1, with no remainder.
Then:
F(x) = (x – 2)(-x2 – x + 1)
#8
Use the factor theorem and synthetic division to decide whether the second polynomial is
a factor of the first for the following:
(a)x^3+6x^2-2x-7; x+1
−1 1
6 −2 −7
−1 −5 7
1 5 −7 0
Since the remainder is 0, x + 1 is a factor of the polynomial.
(b)-2x^3+x^2-63; x+3
−3 −2 1 0 −63
6 −21 63
−2 7 −21 0
Since the remainder is 0, x + 3 is a factor of the polynomial
#9
Sketch the graph of each polynomial function:
(a)f(x)=-x^4+2
The graph of the function looks like this:
(b)f(x)=(x+2)^3-1
The graph of the function looks like this: