Midterm 2 Review
Simplify #1-5
2x2
3
−
x
1) x
5
x
2) 3)
3x
2
3
5x4
2x
5
2 x
+
x 3
1
4)
x+3
4
+
2x
x−1
2
1+x
5) 2
−2
1+x
For #6-13 solve for x.
6)
3x − 1
=4
x+2
7) (7 − x)3 + 4 = 12
2
8)
√
2
3x − 2 − 6 = −4
9) 10 ≤ 2 − x3
10)
11)
√
2x − 1 ≥ 4
2
≤5
x−3
3
12)
3
≤4
2−x
13)
1
≥3
5−x
14) If g(x) is an invertible function, and g(2) = 7, then what is g −1 (7)?
For #15-16 find the inverse. Check each answer by making sure that h−1 ◦ h(x) = x and h ◦ h−1 (x) = x.
√
15) h(x) = 3 2x + 4
4
16) h(x) =
2x
5 − 3x
Use graph transformations to graph #17-19. Make sure to explain each step of the transformation in words.
Also state the domain, range, x-intercept(s) and y-intercept of each.
√
17) f (x) = 2 3 − x + 1
x-int(s)=
have you...labeled the axes with variable names? indicated units? labeled points with coordinates? titled your graph?
y-int=
D=
R=
5
18) f (x) =
√
3
x+2−1
x-int(s)=
have you...labeled the axes with variable names? indicated units? labeled points with coordinates? titled your graph?
y-int=
D=
R=
√
19) f (x) = − 2 x + 2 + 1
x-int(s)=
have you...labeled the axes with variable names? indicated units? labeled points with coordinates? titled your graph?
y-int=
D=
R=
6
20) Graph and state the domain, range, x-intercept, and y-intercept of f (x) = 32 x + 1
x-int(s)=
have you...labeled the axes with variable names? indicated units? labeled points with coordinates? titled your graph?
y-int=
D=
R=
21) Use graph transformations to graph f (x) = −3(x − 2)2 + 3. Make sure to explain each step of the
transformation in words. Also state the domain, range, x-intercepts(s), y-intercept, and vertex.
x-int(s)=
have you...labeled the axes with variable names? indicated units? labeled points with coordinates? titled your graph?
y-int=
D=
R=
vertex=
22) Find the quotient:
12x3 − 13x2 + 9x − 2
x2 − 1
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23) What is the equation of the linear polynomial that passes through the points (2, 4) and (5, 10)?
24) (a) Complete the square: Write f (x) = 3x2 + 6x − 5 in the form α(x + β)2 + γ where α, β, γ ∈ R.
(b) What is the vertex of f (x)?
(c) Find the roots using the form from (a).
For #25-26 determine how many roots the quadratic polynomials have. Solve for these roots using the
quadratic formula (if there are any)
25) 2x2 − 6x + 10
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26) −2x2 + 6x + 2
27) Is (x − 1) a factor of x5 + 3x4 + x3 − x2 − x − 1? Explain.
28) Give an upper and lower bound on the number of roots that the polynomial p(x) = x4 − 5x5 + 3x4 − 1
has and justify your answer.
29) Find a root of f (x) = x4 + x3 − 6x2 − 4x + 8 and verify that it is a root.
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30) Completely factor f (x) = 3x3 − 4x2 − x + 2. (Your answer should be a product of a constant and maybe
some linear and quadratic polynomials that have leading coefficients equal to 1, and such that any of the
quadratics in the product have no roots).
31) Completely factor f (x) = −2x3 + 6x2 − 2x + 6. (Hint: x = 3 is a root.) (Your answer should be a product
of a constant and maybe some linear and quadratic polynomials that have leading coefficients equal to 1,
and such that any of the quadratics in the product have no roots).
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