Precalculus Midterm
Review
1)
Write the following using interval notation:
(a) x ≥ 3
2)
3)
4)
6)
(a)
2x = 3y 2
(d)
9x 2 − y 2 = 9
(c)
0< x≤8
(c)
x 2 + 4 y 2 = 16
(d) − 5 ≤ x < 2
(b)
y = −3 x
(e)
y = x3 − x
Find the domain of each function:
(a)
x
f ( x) = 2
x −9
(d)
f ( x) = 3 x + 2
(b)
3x 2
f ( x) =
x−2
(e)
f ( x) =
(c)
f ( x) = 2 − x
x
x + 2x − 3
2
f
:
g
f ( x) = 3x 2 + x + 1; g ( x) = 3x
For the following functions, find f + g , f − g , f ⋅ g ,
f ( x) = 2 − x; g ( x) = 3x + 1
(b)
Simplify each complex expression:
(a)
(2 − 3i ) + (6 + 8i )
(b)
(−3 + 2i ) − (4 − 4i )
(c)
8(2 − 7i )
(d)
(−3 + 2i )(4 − 4i )
Use the quadratic formula to solve:
(a)
7)
x ≤ −7
Test each equation for symmetry:
(a)
5)
(b)
x 2 − 6 x + 13
(b)
x 2 − 6 x + 10
Graph:
(a)
⎧− 2 x + 3,
f ( x) = ⎨
⎩3x − 2,
x <1
x ≥1
(b)
⎧ x + 3,
f ( x) = ⎨
⎩− 2 x − 3,
x < −2
x ≥ −2
8)
For the following graph, determine the following:
(a)
(b)
(c)
(d)
Domain, Range
Intervals of increasing and decreasing
Min/Max
Intercepts
4.0
3.0
2.0
1.0
−4.0
−3.0
−2.0
−1.0
1.0
2.0
3.0
−1.0
−2.0
−3.0
−4.0
9)
Find the resulting function after all of the transformations below are applied to the
graph of f ( x) = x :
(a)
10)
Shift up 2 units
Reflect x-axis
Reflect y-axis
(b)
Reflect x-axis
Shift right 3 units
Shift down 2 units
(c)
Graph each function (without a calculator):
(a)
f ( x) = x 2 − 1
(b)
f ( x) = − x 2 + 4
(c)
(d)
f ( x) = x − 2 + 3
(f)
f ( x) = 3 − x + 1 − 2
(g)
f ( x) = ( x − 3) 3 − 1
1
f ( x) =
−2
x+4
f ( x) = 2 x + 2 − 3
(h)
f ( x) = e − x +5 + 1
(i)
f ( x) = x + 3 − 5
(j)
f ( x) = − x − 1 + 4
(e)
Shift up 2 units
Reflect y-axis
Shift left 3 units
4.0
5.
11)
Form a polynomial with the given zeros:
(a)
(c)
12)
(c)
x 4 − 3x 2 − 4 ≥ 0
x 3 − 3x 2 − 6 x + 8
x 4 − 4 x 3 + 9 x 2 − 20 x + 20
(b)
(d)
x 3 − x 2 − 10 x − 8
x 4 + 6 x 3 + 11x 2 + 12 x + 18
x 4 + 2x 2 − 8 = 0
3x 4 − 4 x 3 + 4 x 2 − 4 x + 1
(b)
x 3 − x 2 − 8 x + 12 = 0
f ( x) = 8 x 3 − 3x 2 + x + 4; g ( x) = x − 1
f ( x) = 2 x 3 + 8 x 2 − 5 x + 5; g ( x) = x − 2
f ( x) = x( x + 2)( x − 4)
(b)
f ( x) = ( x − 1) 2 (3 x + 2)( x 2 + 4)
(b)
f ( x) = 3x + 2; g ( x) = 2 x 2 − 1
Find f ( g ( x)) :
(a)
18)
x ( x − 7) > 8
For the following, determine the degree of each polynomial, the zeros, and whether
each zero crosses or touches the x-axis,
(a)
17)
(b)
Determine whether g(x) is a factor of f(x) using synthetic division:
(a)
(b)
16)
x 2 + 7 x < −12
Solve each equation for the complex zeros:
(a)
(c)
15)
Zeros: -2, 2, 3
Zeros: 2 (mult 2), 3, 5
Find the zeros of each polynomial function:
(a)
(c)
14)
(b)
(d)
Solve each polynomial inequality:
(a)
13)
Zeros: -1, 1, 3
Zeros: -4, 0, 2
f ( x) = 2 x;
g ( x) = 3x 2 + 1
Find the horizontal, vertical, and slant asymptotes of the functions below.
(a)
2x3 − x + 7
7x3 −1
(b)
3x + x 2 − 8 x + 2
x2 −1
(c)
3x 2 − 5
x+5
(d)
3x + 8
x−9
19)
20)
21)
Determine the inverse:
(a)
{(4,−6), (−1,0), (5,−3)}
(b)
f ( x) = 4 x + 2
(c)
f ( x) =
5
x −3
(d)
f ( x) = ( x − 4) 3 + 1
Solve each equation:
9−x =
(f)
9 2 x = 27
(d)
(g)
ln(3 x + 1) = 3
(h)
log 5 7 − x = 2
(i)
log 7 ( x − 48) + log 7 x = 2
(j)
log( x + 4) − log x = 2
(k)
e 2 x − 6 = 23
(l)
7 x + 4 = 39−5 x
2
=
(b)
8x
(e)
4x = 8
1
3
(c)
⎛1⎞
⎜ ⎟
⎝2⎠
1− x
=4
Rewrite the following expressions as logarithmic expressions.
6a = b
b)
2 5 = 32
Rewrite the following expressions as exponential expressions.
a)
23)
1
2
2 2 x +1 = 4
a)
22)
−2 x
(a)
log 7 a = b
b)
ln x = 3
Classify each expression as growth or decay. Then indicate the rate of growth or
decay.
a)
P (t ) = 4(1.06)t
b)
f ( x) = 109(0.873) x
c)
y = 0.5(1.12) x
d)
1
s (t ) = (1.0065) x
2
24)
25)
Write the expressions below as the sum and/or difference of logs.
a)
log x + 2 3 x − 4
b)
ln
c)
⎡ ( x 2 − 1)( x 3 ) ⎤
log ⎢
⎥
4
⎣ 3x − x ⎦
4( x + 2)
3 x ( x − 7)
5
Write the expressions below as a single log.
3
a)
3 log x − log y + log z
2
b)
log(2 x + 3) − (log x + log 2)
c)
1
log 6 80 + log 6 ( x 2 + 5) − log 6 (7 − 2 x)
2
One Last Note: If this review leaves you wanting more, then you can always go
back and redo old homework assignments and old tests and quizzes!!!
You can do this – but you have to do what?
WORK AT IT.
WORK AT IT SOME MORE.
WHEN YOU THINK YOU HAVE WORKED AT THIS ENOUGH, THAT’S
WHEN YOU WORK EVEN MORE.
This is a chance for you to see what you can push yourself to overcome mentally. This
is a chance to prove that you can take an unprecedented situation and turn it into
something positive for yourself!
LET’S DO IT!!!!!!!!!!