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MATH 130 Answers to Test 1 (Form B) Autumn 2007
- Solve the given equation.
(i)
x + 6 x^2 โ 3 x
x
x x โ 3
x(x โ 3) x + 6 x(x โ 3) โ x(x โ 3)
x = x(x โ 3) x x โ 3
x + 2 โ (x โ 3) = x(x); x + 6 โ x + 3 = x^2 ; 9 = x^2 ; x = 3 or x = โ3.
Check : If x = 3, x + 6 x^2 โ 3 x
x
is undefined.
If x = โ3, x + 6 x^2 โ 3 x
x
(โ3)^2 โ 3(โ3)
x x โ 3
= 0.5. Answer: x = โ 3
(ii) Solve
y + 6 = y + 4. y + 6 = (y + 4)(y + 4) = y^2 + 4y + 4y + 16; 0 = y^2 + 8y + 10 = (y + 2)(y + 5). So y = โ 2 or y = โ5. If y = โ2,
y + 6 =
โ2 + 6 = 2 and y + 4 = โ2 + 4 = 2. If y = โ5,
y + 6 =
โ5 + 6 = 1 but y + 4 = โ5 + 4 = โ1. Answer: y = โ 2
- Solve the inequality 2(x โ 1) โ 3(x + 1) โค 5. Give your answer using interval notation.
2 x โ 2 โ 3 x โ 3 โค 5; โx โค 10; x โฅ โ10. Answer: [โ 10 , โ)
(over) Typeset using AMS-TEX.
- Let f (x) = x^2 โ 4 x.
(i) Find f (โ2). Simplify your answer as much as possible. (โ2)^2 โ 4(โ2) Answer: 12
(ii) Find f (x + h). Simplify your answer as much as possible. (x + h)^2 โ 4(x + h) = (x + h)(x + h) โ 4 x โ 4 h = x^2 + xh + hx + h^2 โ 4 x + 4h Answer: x^2 + 2xh + h^2 โ 4 x โ 4 h
(iii) Find f (x + h) โ f (x) h
. Simplify your answer as much as possible. x^2 + 2xh + h^2 โ 4 x โ 4 h โ x^2 + 4x h
2 xh + h^2 โ 4 h h Answer: 2x + h โ 4
- Find the domain of the given function.
(i) h(x) =
3 โ 7 x.
3 โ 7 x โฅ 0; 3/ 7 โฅ x. Answer: (โโ, 3 /7]
(ii) g(x) = x + 1 x^2 โ 4
. Answer: All numbers except ยฑ2.
- Let F (x) = x^2 โ 2 x and G(x) = x โ 3.
(i) Find (F G)(x). Simplify your answer as much as possible.
(x^2 โ 2 x)(x โ 3) = x^3 โ 3 x^2 โ 2 x^2 + 6x Answer: x^3 โ 5 x^2 + 6x
(ii) Find (F โฆ G)(x). Simplify your answer as much as possible.
F (G(x)) = F (x โ 3) = (x โ 3)^2 โ 2(x โ 3) = x^2 โ 6 x + 9 โ 2 x + 6 Answer: x^2 โ 8 x + 15
- Find functions f (x) and g(x), neither of which is the identity function, such that (f โฆ g)(x) =
x + 2.
f (x):
x g(x): x + 2
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- A lumber company owns a wooded area that is in the shape of a rectangle that measures 2 miles by 3 miles. They are planning to clear a uniform strip along the outer edges of the forest, so that 3 square miles of forest remain.
3 miles
2 miles
x
x
x x
(i) Write down a quadratic equation for the width of the sttrip. Let the width of the strip be x. Then the area of the forest that remains uncut is (2 โ 2 x)(3 โ 2 x). So we want (2 โ 2 x)(3 โ 2 x) = 3; 6 โ 4 x โ 6 x + 4x^2 = 3; Answer: 4x^2 โ 10 x + 3 = 0
(ii) Solve the quadratic equation to find the width of the strip. Give your answer to the nearest 0.01 mile.
x =
(โ10)^2 โ 4(4)(3)
โ 2 .15 or 0. 35
The width of the strip cannot be more than 2/2 = 1 mile. Answer: 0.35 mile
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