# PSAT Math Quiz: Functions

Content Review:

Answer 1

**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.

Answer 2

**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).

Answer 3

**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.

Answer 4

**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.