12 multiple-choice questions, progressively harder.
Use the function
G(x)={x+1if x<0x2if 0≤x≤27−xif x>2G(x) = \begin{cases} x + 1 & \text{if } x < 0 \\ x^2 & \text{if } 0 \le x \le 2 \\ 7 - x & \text{if } x > 2 \end{cases}G(x)=⎩⎨⎧x+1x27−xif x<0if 0≤x≤2if x>2
to evaluate G(−2)G(-2)G(−2).
Solution
Correct answer: B
Since −2<0-2 < 0−2<0, the first piece x+1x + 1x+1 applies.
G(−2)=−2+1=−1G(-2) = -2 + 1 = -1G(−2)=−2+1=−1
For what value of ccc does the function
f(x)={x+cif x<13xif x≥1f(x) = \begin{cases} x + c & \text{if } x < 1 \\ 3x & \text{if } x \ge 1 \end{cases}f(x)={x+c3xif x<1if x≥1
connect (no jump) at x=1x = 1x=1?
Correct answer: D
Connecting at x=1x = 1x=1 means both pieces give the same value there. Set the left value equal to the right value: 1+c=3(1)1 + c = 3(1)1+c=3(1).
1+c=3⇒c=21 + c = 3 \Rightarrow c = 21+c=3⇒c=2
What is the range of
Correct answer: C
Take pieces separately. The third piece 7−x7 - x7−x on x>2x > 2x>2 already produces every value below 555, and no piece reaches 555 or above.
(−∞,1)∪[0,4]∪(−∞,5)=(−∞,5)(-\infty, 1) \cup [0, 4] \cup (-\infty, 5) = (-\infty, 5)(−∞,1)∪[0,4]∪(−∞,5)=(−∞,5)
So the range is y<5y < 5y<5.
For the function
how many inputs xxx satisfy G(x)=4G(x) = 4G(x)=4?
Set each piece equal to 444 on its interval. The middle gives x2=4⇒x=2x^2 = 4 \Rightarrow x = 2x2=4⇒x=2 (valid), the third gives 7−x=4⇒x=37 - x = 4 \Rightarrow x = 37−x=4⇒x=3 (valid); the first would need x=3x = 3x=3, rejected since it requires x<0x < 0x<0.
x=2 and x=3x = 2 \ \text{and} \ x = 3x=2 and x=3
So two inputs map to 444.
A cinema charges by age aaa in years, in dollars:
{0if a<38if 3≤a<1312if 13≤a<656if a≥65\begin{cases} 0 & \text{if } a < 3 \\ 8 & \text{if } 3 \le a < 13 \\ 12 & \text{if } 13 \le a < 65 \\ 6 & \text{if } a \ge 65 \end{cases}⎩⎨⎧08126if a<3if 3≤a<13if 13≤a<65if a≥65
What is the price at exactly age 131313?
Correct answer: A
Age 131313 satisfies 13≤a<6513 \le a < 6513≤a<65, so that tier applies.
price=12 dollars\text{price} = 12 \text{ dollars}price=12 dollars
The boundary 131313 leaves the child tier and enters the adult tier.
What is the price at exactly age 333?
Age 333 satisfies 3≤a<133 \le a < 133≤a<13, so that tier applies.
price=8 dollars\text{price} = 8 \text{ dollars}price=8 dollars
The boundary 333 leaves the free tier.
A graph has a closed (filled) dot at (2,5)(2, 5)(2,5) and an open (hollow) dot at (2,1)(2, 1)(2,1). What is f(2)f(2)f(2)?
The closed (filled) dot marks the point the graph actually contains at x=2x = 2x=2.
f(2)=5f(2) = 5f(2)=5
The open dot at (2,1)(2, 1)(2,1) belongs to the other piece and is not the value.
At x=4x = 4x=4, a graph's left piece ends with a hollow dot at (4,3)(4, 3)(4,3) and its right piece begins with a filled dot at (4,3)(4, 3)(4,3). What happens at x=4x = 4x=4?
The hollow and filled dots land on the same point, so the hole is filled and the graph is unbroken there.
f(4)=3f(4) = 3f(4)=3
Same height from both pieces means the graph connects.
f(x)={∣x∣if x<2x−2if x≥2f(x) = \begin{cases} \lvert x \rvert & \text{if } x < 2 \\ x - 2 & \text{if } x \ge 2 \end{cases}f(x)={∣x∣x−2if x<2if x≥2
to evaluate f(−3)f(-3)f(−3).
Since −3<2-3 < 2−3<2, the first piece ∣x∣\lvert x \rvert∣x∣ applies.
f(−3)=∣−3∣=3f(-3) = \lvert -3 \rvert = 3f(−3)=∣−3∣=3
to evaluate f(2)f(2)f(2).
The boundary 222 satisfies x≥2x \ge 2x≥2, so the second piece x−2x - 2x−2 applies.
f(2)=2−2=0f(2) = 2 - 2 = 0f(2)=2−2=0
The first piece would give ∣2∣=2\lvert 2 \rvert = 2∣2∣=2, but it needs x<2x < 2x<2.
Does the graph of
jump or connect at x=2x = 2x=2?
Compare both pieces at 222: the left piece ∣x∣\lvert x \rvert∣x∣ heads to 222, the right piece x−2x - 2x−2 is 000.
2≠02 \ne 02=0
Different heights, so the graph jumps.
Solve f(x)=0f(x) = 0f(x)=0 for the function
f(x)={2x+6if x<−1x−4if x≥−1f(x) = \begin{cases} 2x + 6 & \text{if } x < -1 \\ x - 4 & \text{if } x \ge -1 \end{cases}f(x)={2x+6x−4if x<−1if x≥−1
Set each piece to 000 on its interval. First: 2x+6=0⇒x=−32x + 6 = 0 \Rightarrow x = -32x+6=0⇒x=−3 (valid, −3<−1-3 < -1−3<−1). Second: x−4=0⇒x=4x - 4 = 0 \Rightarrow x = 4x−4=0⇒x=4 (valid, 4≥−14 \ge -14≥−1).
x=−3 or x=4x = -3 \ \text{or} \ x = 4x=−3 or x=4
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