12 multiple-choice questions, progressively harder.
Use the function
C(x)={−xif x<02if 0≤x<3x−1if x≥3C(x) = \begin{cases} -x & \text{if } x < 0 \\ 2 & \text{if } 0 \le x < 3 \\ x - 1 & \text{if } x \ge 3 \end{cases}C(x)=⎩⎨⎧−x2x−1if x<0if 0≤x<3if x≥3
to evaluate C(−4)C(-4)C(−4).
Solution
Correct answer: D
Since −4<0-4 < 0−4<0, the first piece −x-x−x applies.
C(−4)=−(−4)=4C(-4) = -(-4) = 4C(−4)=−(−4)=4
Negating a negative gives a positive result.
B(x)={x2if x≤12x−1if x>1B(x) = \begin{cases} x^2 & \text{if } x \le 1 \\ 2x - 1 & \text{if } x > 1 \end{cases}B(x)={x22x−1if x≤1if x>1
to evaluate B(1)B(1)B(1).
Correct answer: B
The boundary 111 satisfies x≤1x \le 1x≤1, so the first piece x2x^2x2 applies.
B(1)=12=1B(1) = 1^2 = 1B(1)=12=1
The second piece also heads to 111 here, so the graph connects, but the value is read from the piece that owns the point.
to evaluate B(3)B(3)B(3).
Since 3>13 > 13>1, the second piece 2x−12x - 12x−1 applies.
B(3)=2(3)−1=5B(3) = 2(3) - 1 = 5B(3)=2(3)−1=5
The value 9=329 = 3^29=32 uses the wrong piece.
to evaluate B(−2)B(-2)B(−2).
Correct answer: A
Since −2≤1-2 \le 1−2≤1, the first piece x2x^2x2 applies.
B(−2)=(−2)2=4B(-2) = (-2)^2 = 4B(−2)=(−2)2=4
Squaring a negative gives a positive result.
A(x)={x+4if x<23x−1if x≥2A(x) = \begin{cases} x + 4 & \text{if } x < 2 \\ 3x - 1 & \text{if } x \ge 2 \end{cases}A(x)={x+43x−1if x<2if x≥2
to evaluate A(2)A(2)A(2).
Correct answer: C
The boundary 222 satisfies x≥2x \ge 2x≥2, so the second piece 3x−13x - 13x−1 applies.
A(2)=3(2)−1=5A(2) = 3(2) - 1 = 5A(2)=3(2)−1=5
The value 6=2+46 = 2 + 46=2+4 uses the first piece, which excludes 222.
Does the graph of
jump or connect at x=3x = 3x=3?
Evaluate both pieces at 333. The middle piece is 222 and the right piece is 3−1=23 - 1 = 23−1=2.
2=22 = 22=2
Same height, so the graph connects at x=3x = 3x=3.
jump or connect at x=1x = 1x=1?
Evaluate both pieces at 111: 12=11^2 = 112=1 and 2(1)−1=12(1) - 1 = 12(1)−1=1.
1=11 = 11=1
Equal heights, so the graph connects, though with a corner because the slopes differ.
jump or connect at x=2x = 2x=2?
Evaluate both pieces at 222: the left piece heads to 2+4=62 + 4 = 62+4=6, the right piece is 3(2)−1=53(2) - 1 = 53(2)−1=5.
6≠56 \ne 56=5
Unequal heights, so the graph jumps.
What is the domain of
The conditions x<0x < 0x<0, 0≤x<30 \le x < 30≤x<3, and x≥3x \ge 3x≥3 cover every real number with no gaps.
{x<0}∪{0≤x<3}∪{x≥3}=R\{x < 0\} \cup \{0 \le x < 3\} \cup \{x \ge 3\} = \mathbb{R}{x<0}∪{0≤x<3}∪{x≥3}=R
So the domain is all real numbers.
A shipping charge in dollars is
C(w)={4if 0<w≤17if 1<w≤310if w>3C(w) = \begin{cases} 4 & \text{if } 0 < w \le 1 \\ 7 & \text{if } 1 < w \le 3 \\ 10 & \text{if } w > 3 \end{cases}C(w)=⎩⎨⎧4710if 0<w≤1if 1<w≤3if w>3
What is the charge for w=3w = 3w=3 pounds?
The weight 333 satisfies 1<w≤31 < w \le 31<w≤3, so the second tier applies.
C(3)=7 dollarsC(3) = 7 \text{ dollars}C(3)=7 dollars
The boundary 333 belongs to the tier whose condition includes it.
What is the charge for w=3.5w = 3.5w=3.5 pounds?
The weight 3.53.53.5 satisfies w>3w > 3w>3, so the third tier applies.
C(3.5)=10 dollarsC(3.5) = 10 \text{ dollars}C(3.5)=10 dollars
What is the range of
Take pieces separately. On x≤1x \le 1x≤1, x2x^2x2 reaches every value from 000 upward; on x>1x > 1x>1, 2x−1>12x - 1 > 12x−1>1.
[0,∞)∪(1,∞)=[0,∞)[0, \infty) \cup (1, \infty) = [0, \infty)[0,∞)∪(1,∞)=[0,∞)
So the range is y≥0y \ge 0y≥0.
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