12 multiple-choice questions, progressively harder.
The graph of a function fff is best described as which collection of points?
Solution
Correct answer: B
The graph pairs each input with its output, plotting the input as the horizontal coordinate and the output as the vertical coordinate.
graph of f={(x,f(x))}\text{graph of } f = \{(x, f(x))\}graph of f={(x,f(x))}
Every point on the graph has an input as its first coordinate and the matching output as its second.
A function fff satisfies f(2)=5f(2) = 5f(2)=5. Which point must lie on the graph of fff?
A point (a,b)(a, b)(a,b) is on the graph exactly when b=f(a)b = f(a)b=f(a): the input is first, the output second.
f(2)=5 ⇒ (2,5) is on the graphf(2) = 5 \;\Rightarrow\; (2, 5) \text{ is on the graph}f(2)=5⇒(2,5) is on the graph
The input 222 is the horizontal coordinate and the output 555 is the vertical coordinate.
For f(x)=x+4f(x) = x + 4f(x)=x+4, is the point (1,5)(1, 5)(1,5) on the graph?
Correct answer: A
Check whether the output at the input 111 equals the point's second coordinate.
f(1)=1+4=5f(1) = 1 + 4 = 5f(1)=1+4=5
Since f(1)=5f(1) = 5f(1)=5 matches, the point (1,5)(1, 5)(1,5) is on the graph.
The graph of ggg passes through (0,1)(0, 1)(0,1), (1,3)(1, 3)(1,3), (2,5)(2, 5)(2,5), and (3,7)(3, 7)(3,7). What is g(2)g(2)g(2)?
Correct answer: D
Reading g(2)g(2)g(2) means finding the output at input 222, so locate the point whose first coordinate is 222.
(2,5) is on the graph ⇒ g(2)=5(2, 5) \text{ is on the graph} \;\Rightarrow\; g(2) = 5(2,5) is on the graph⇒g(2)=5
The second coordinate, 555, is the output.
A function hhh has h(1)=4h(1) = 4h(1)=4, h(2)=1h(2) = 1h(2)=1, h(3)=0h(3) = 0h(3)=0, and h(4)=1h(4) = 1h(4)=1. What is h(3)h(3)h(3)?
Correct answer: C
Read the value paired with the input 333 directly from the given data.
h(3)=0h(3) = 0h(3)=0
The input is 333 and its listed output is 000.
The graph of fff is shown. What is f(0)f(0)f(0)?
To read f(0)f(0)f(0), start at x=0x = 0x=0 on the horizontal axis and go straight to the curve.
f(0)=−3f(0) = -3f(0)=−3
The curve meets the vertical axis at height −3-3−3, so that is the output at x=0x = 0x=0.
The straight-line graph of fff is shown. What is f(2)f(2)f(2)?
Start at x=2x = 2x=2 on the horizontal axis and move to the line.
f(2)=0f(2) = 0f(2)=0
The line passes through (2,0)(2, 0)(2,0), so the output at x=2x = 2x=2 is 000 (this is also the x-intercept).
A graph is drawn only for inputs from x=−2x = -2x=−2 to x=5x = 5x=5. What is its domain?
The domain is the horizontal spread of the graph, from its leftmost input to its rightmost.
−2≤x≤5-2 \le x \le 5−2≤x≤5
The inputs stop at the two endpoints, so the domain stops there too.
On the graph of a function, the value f(a)f(a)f(a) is read as which of these?
The output f(a)f(a)f(a) is the second coordinate of the point (a,f(a))(a, f(a))(a,f(a)), measured against the vertical axis.
f(a)=height of the graph above x=af(a) = \text{height of the graph above } x = af(a)=height of the graph above x=a
The horizontal axis only tells you the input aaa.
At the bottom of a valley on a graph, the output is smaller than at any nearby input. This point is called what?
A point lower than all nearby points is the smallest output in its neighborhood.
lowest nearby output ⇒ minimum\text{lowest nearby output} \;\Rightarrow\; \text{minimum}lowest nearby output⇒minimum
The top of a hill would be a maximum instead.
A function ppp has p(0)=2p(0) = 2p(0)=2, p(1)=0p(1) = 0p(1)=0, p(2)=2p(2) = 2p(2)=2, and p(3)=6p(3) = 6p(3)=6. Which input is an x-intercept of its graph?
An x-intercept is an input whose output is 000.
p(1)=0 ⇒ x=1p(1) = 0 \;\Rightarrow\; x = 1p(1)=0⇒x=1
The point (1,0)(1, 0)(1,0) sits on the horizontal axis.
Solving f(x)=0f(x) = 0f(x)=0 finds which feature of the graph?
Setting the output to 000 locates the points on the horizontal axis.
f(x)=0 ⇒ x-interceptsf(x) = 0 \;\Rightarrow\; \text{x-intercepts}f(x)=0⇒x-intercepts
These inputs are the zeros of the function.
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