Math 1090-2 Midterm-2 Practice
Name and Unid:
1. Solve the following quadratic equations using factoring AND completing the square AND quadratic
formula:
(a) x2 + 8x = 0;
(b) 2x2 + x = 6;
(c) x2 + 408x + 2015 = 0;
(d) −x2 − 3x + 10 = 0;
2. Solve the following rational equations:
1
1
−
= 1;
(a)
x−3 x+2
2
(b) 1 + x =
;
3−x
3. Determine the vertex, axis of symmetry, maximum value or minimum value, x-intercept(s) if any and
y-intercept. Sketch the graph in the figure below.
(a) y = 2x2 + 4x − 16;
(b) y = −3x2 + 12x + 1;
5
4
3
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
-4
-5
4. Suppose a company provides a kind of products with the fixed cost of $500. The variable cost 2x + 400
per product and the selling price of a product is 1000 − x when the company provides x amounts of
this product.
(a) What’s the Cost Function?
(b) How many should this company produce if they want to minimize the cost?
(c) What’s the Revenue Function?
(d) How many should they produce if they want to maximize revenue?
(e) What’s the Profit Function?
(f) How many should they produce if they want to maximize profit?
(g) Should they maximize profit or revenue?
(h) How many should they produce if they want to break even?
5. Find the vertical asymptotes and horizontal asymptotes of the following rational equations if any:
2x3 + 3x4
;
3x4 + x2 + 1
2x + 1
(b)
;
x
3x5
(c) 2
x +1
(a)
6. Given the base function f (x) = x2 , draw the graph of it below and draw the intermediate steps in dashed
lines to show how you can reach the graph of g(x) = −2x2 + 4x + 1.
5
4
3
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
-1
-2
-3
-4
-5
7. Given 3 functions
√
1
f (x) = 2 2 x − 1, g(x) = x2 + 1, h(x) = x2 ;
4
(a) What’s the domain of those 3 functions? Range?
(b) Compute f ◦ g;
(c) Compute f ◦ h;
(d) Compute g ◦ f ;
(e) Compute h(f (x));
(f) Compute f (g(h(x)));
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5
(g) Compute f · g;
(h) What conclusion can you make on the relationship between f (x) and g(x)? Why?
8. Find the inverse functions of the following functions and specify the domains.
r
x+2
(a) f (x) = 5
x+3
(b) g(x) = e2x
(c) h(x) = x
x3 − 5
x3 + 1
(e) j(x) = ln(3x − 1)
(d) i(x) =
9. List the 8 properties of the logarithms.
10. Solve the following equations:
(a) ex = 1;
(b) 10x+2 + 10x−1 = 21;
(c) 22x − 3 · 2x − 4 = 0;
(d) log2 x + log4 x = 1;
(e) log x2 + log(x − 3) = 4;
(f) ln 2x + ln(x − 1) = 3;
(g) log5 (4 − x) + log5 (x − 8) = 2;
(h) ln x3 = (ln x)3
11. The figure below is the graph for these functions
ex , e2x , e3x , e−x , e−2x , e−3x .
Match the graph with the functions. What’s the intersection point in the figure?
20
18
16
14
y
12
10
8
6
4
2
0
-1
-0.5
0
x
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0.5
1
12. The figure below is the graph for these functions
log2 x, log3 x, log4 x, log 21 x, log 31 x, log 14 x.
Match the graph with the functions. What’s the intersection point in the figure?
6
4
y
2
0
-2
-4
-6
0
0.5
1
x
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1.5
2