12 multiple-choice questions, progressively harder.
The graph of ggg passes through (−1,2)(-1, 2)(−1,2), (0,−1)(0, -1)(0,−1), (1,−2)(1, -2)(1,−2), (2,−1)(2, -1)(2,−1), and (3,2)(3, 2)(3,2). For which inputs does g(x)=2g(x) = 2g(x)=2?
Solution
Correct answer: D
Solving g(x)=2g(x) = 2g(x)=2 means finding every input paired with output 222.
g(−1)=2andg(3)=2g(-1) = 2 \quad \text{and} \quad g(3) = 2g(−1)=2andg(3)=2
Two different inputs share the output 222, so there are two solutions.
The graph of fff is shown. What is the maximum value of fff?
The maximum value is the greatest height the curve reaches, at the top of the hill.
highest point (1,4) ⇒ maximum 4\text{highest point } (1, 4) \;\Rightarrow\; \text{maximum } 4highest point (1,4)⇒maximum 4
The maximum value is the output 444, reached at x=1x = 1x=1.
For f(x)=3x+6f(x) = 3x + 6f(x)=3x+6, what is the x-intercept of its graph?
Correct answer: B
The x-intercept has output 000, so solve f(x)=0f(x) = 0f(x)=0.
3x+6=0 ⇒ x=−23x + 6 = 0 \;\Rightarrow\; x = -23x+6=0⇒x=−2
The x-intercept is the point (−2,0)(-2, 0)(−2,0).
The graph of a function runs off to the left and right forever, dipping to a lowest point at (2,−5)(2, -5)(2,−5). What is its range?
Correct answer: C
The curve reaches its lowest output at −5-5−5 and climbs without bound on both sides, so every height from −5-5−5 up is covered.
range : y≥−5\text{range} \colon \ y \ge -5range: y≥−5
No output falls below the minimum −5-5−5.
The graph of ggg passes through (−2,0)(-2, 0)(−2,0), (0,−4)(0, -4)(0,−4), and (2,0)(2, 0)(2,0), opening upward. What is g(0)g(0)g(0)?
Correct answer: A
Read the output at input 000 from the given point.
g(0)=−4g(0) = -4g(0)=−4
The point (0,−4)(0, -4)(0,−4) is the y-intercept.
The graph of fff is shown. On which interval is fff increasing?
The curve falls until the turning point at x=1x = 1x=1, then climbs.
rises for x>1 ⇒ increasing for x>1\text{rises for } x > 1 \;\Rightarrow\; \text{increasing for } x > 1rises for x>1⇒increasing for x>1
To the left of x=1x = 1x=1 it is decreasing.
For f(x)=x2−9f(x) = x^2 - 9f(x)=x2−9, at which two inputs does the graph cross the x-axis?
Set the output to 000 and solve.
x2−9=0 ⇒ x=−3 or x=3x^2 - 9 = 0 \;\Rightarrow\; x = -3 \text{ or } x = 3x2−9=0⇒x=−3 or x=3
The x-intercepts are (−3,0)(-3, 0)(−3,0) and (3,0)(3, 0)(3,0).
A curve is drawn only between endpoints (−4,−1)(-4, -1)(−4,−1) and (2,−1)(2, -1)(2,−1). What is its domain?
The domain is the horizontal spread, from the leftmost input to the rightmost.
−4≤x≤2-4 \le x \le 2−4≤x≤2
The shared height −1-1−1 is about the range, not the domain.
For f(x)=−x2+4f(x) = -x^2 + 4f(x)=−x2+4, what is the maximum value of the function?
The rule opens downward, so its highest point is the vertex at x=0x = 0x=0.
f(0)=−0+4=4f(0) = -0 + 4 = 4f(0)=−0+4=4
Every other output is smaller, so the maximum value is 444.
If (a,0)(a, 0)(a,0) is a point on the graph of fff, then aaa is which of these?
A point with second coordinate 000 lies on the horizontal axis.
(a,0) on the graph ⇒ f(a)=0(a, 0) \text{ on the graph} \;\Rightarrow\; f(a) = 0(a,0) on the graph⇒f(a)=0
So aaa is a zero of fff, an x-intercept.
A function qqq has q(−1)=3q(-1) = 3q(−1)=3, q(0)=0q(0) = 0q(0)=0, q(1)=−1q(1) = -1q(1)=−1, q(2)=0q(2) = 0q(2)=0, and q(3)=3q(3) = 3q(3)=3. How many solutions does q(x)=0q(x) = 0q(x)=0 have?
Count the inputs whose output is 000.
q(0)=0andq(2)=0q(0) = 0 \quad \text{and} \quad q(2) = 0q(0)=0andq(2)=0
Two inputs give output 000, so there are two x-intercepts.
The graph of fff rises from left to right for all xxx. What can you say about f(1)f(1)f(1) and f(5)f(5)f(5)?
An increasing graph gives larger outputs for larger inputs, and 5>15 > 15>1.
1<5 ⇒ f(1)<f(5)1 < 5 \;\Rightarrow\; f(1) < f(5)1<5⇒f(1)<f(5)
Moving right always raises the height on an increasing graph.
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