PSAT Math Quiz: Functions

Content Review:

Question 1
The figure shows the graph of r(x). Which of the following cannot be a value of x for which r(x) = 0 ?
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A. -3
B. -2
C. 2
D. 5

A: The notation r(x) = 0 means that the function is crossing the x-axis (has a y-value of 0), so look for the x-intercepts. The function r(x) intersects the x-axis at x = -2, 2, and 5. Because B, C, and D are all possible values, the correct answer is A.

Question 2
A construction company plans to build a long row of houses, each with a certain number of brown shingles on its roof. The number of brown shingles on a house’s roof, f (h), is a function of its house number, h. The first few houses are shown here. If only brown shingles are used, how many shingles will the seventh house have?
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A. 7
B. 27
C. 46
D. 60

 D: Identifying the type of relationship between h and f(h) is key to solving this problem. The f(h)  values (shingle counts) are increasing at a variable rate, which rules out a linear function. The increase in f(h) is not significant with each increase in h, so exponential is not likely. A quadratic relationship is a good bet. Start with f(h) = h^2. That gives f(1) = 1^2 = 2, f(2) = 2^2 = 4, and f(3) = 3^2 = 9. Obviously this doesn’t match the pattern, so ask yourself how much more f(h) needs to increase after squaring h; you’ll see you need to add 11 to h^2 + 11 accurately depicts the relationship between h and f(h). You’re asked for the shingle count for the seventh house, so put 7 into your function: f(7) = 7^2 + 11 = 49 + 11 = 60, which matches (D).

Question 3
The function f(x) = (x − 4)² is shown here. Which of the following correctly depicts the transformation g(x) = (–x + 2)+ 3?
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A. psat math
B. psat math
C. psat math
D. psat math

C: Take each transformation one at a time. The negative sign inside the parentheses indicates a horizontal reflection (across the y-axis), so the parabola should still open up. Choice B has a vertical reflection (across the x-axis) and opens down, so you can eliminate it. The +3 outside the parentheses shifts f(x) up 3 units; D does not contain this component, so eliminate it as well. the +2 inside the parentheses is tricky: it means a left shift of 2 units, but it might be tempting to think you need to add 6 (and therefore move 6 units to the left) to get from xI – 4 to x + 2. Don’t be fooled by this. Overall the graph will shift up 3 units, shift left 2 units, then reflect over the y-axis. Algebraically, your function would be g(x) = (-x – 2)^2 + 3 (not g(x) = (-x + 2)^2 + 3); this and the graphical analysis correspond to C, the correct answer. 

Question 4
If f (x) = –x + 5 and g(x) = x2, which of the following is not in the range of f (g(x)) ?
A. -11
B. 0
C. 1
D. 9

D: When dealing with a composition, the range of the inner function becomes the domain of the outer function, which in turn produces the range of the composition. In the composition f(g(x)), the function g(x) = x^2is the inner function. Every value of x, when substituted into this function, will result in a nonnegative value (because of the square on x). This means the smallest possible range value of g(x) is 0. Now look at f(x). Substituting large positive values of x in the function will result in large negative numbers. Consequently, substituting the smallest value from the range of g, which is 0, results in the largest range value for the composition, which is -0 + 5 = 5. Because 9 > 5, it is not in the range of f(g(x)), making D correct.

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