Comparing Different Function Representations, #2 (Grades 11-12)

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Comparing Different Function Representations, #2

1. 
If [math]f(x)[/math] is the exponential function shown in the following graph, and [math]g(x) = 100^x[/math], which of the following is true? Assume that the scale of both axes on the graph is one unit.
Graph - Exponent Function y=2^x
  1. The y-intercepts are the same.
  2. The y-intercept of [math]g(x)[/math] is greater than that of [math]f(x)[/math].
  3. The y-intercept of [math]f(x)[/math] is greater than that of [math]g(x)[/math].
  4. The function [math]g(x)[/math] has no y-intercept.
2. 
If [math]f(x) = (x+3)(x-1)(x-2)[/math] and [math]g(x)[/math] is a polynomial function, some of whose values are given in the table below, which of the following values are zeros of both functions?

[math] \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \mathbf{g(x)} \ \ \ \ \ \ \ \ \ [/math]
[math] -6 [/math][math] 0 [/math]
[math] -5 [/math][math] 10 [/math]
[math] -4 [/math][math] 6 [/math]
[math] -3 [/math][math] 0 [/math]
[math] -2 [/math][math] -2 [/math]
[math] -1 [/math][math] 0 [/math]
[math] 0 [/math][math] 0 [/math]
[math] 1 [/math][math] -14 [/math]
[math] 2 [/math][math] -27 [/math]
  1. 1
  2. 0
  3. -3
  4. -6
3. 
Given the function [math]f(x) = 1/10 (x+4)(x+2)(x-1)(x-5)[/math], and another polynomial function [math]g(x)[/math], some of whose values are listed in the following table, which of the statements concerning these two functions is true?

[math] \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \mathbf{g(x)} \ \ \ \ \ \ \ \ \ [/math]
[math] -4 [/math][math] 20 [/math]
[math] -3.5 [/math][math] 18.5 [/math]
[math] -3 [/math][math] 15.1 [/math]
[math] -2.5 [/math][math] 12.2 [/math]
[math] -2 [/math][math] 14.8 [/math]
  1. Both functions have a local minimum on the interval [math](-4,-2)[/math].
  2. Both functions have a local minimum at [math]x = -2.[/math]
  3. [math]f(x)[/math] has a local minimum on the interval [math](-4,-2)[/math], and [math]g(x)[/math] has a local maximum on this interval.
  4. [math]g(x)[/math] has a local minimum on the interval [math](-4,-2)[/math], and [math]f(x)[/math] has a local maximum on this interval.
4. 
If [math]f(x)[/math] is an exponential function, some of whose values are listed in the table, and [math]g(x) = 5^(-x)[/math], which of the following statements is true?

[math] \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \ [/math]
[math] -1 [/math][math] 11 [/math]
[math] 0 [/math][math] 3 [/math]
[math] 1 [/math][math] -1 [/math]
[math] 10 [/math][math] -3 [/math]
[math] 100 [/math][math] -4.75 [/math]
[math] 1000 [/math][math] -4.99 [/math]
  1. If both functions were graphed, the horizontal asymptote of [math]f(x)[/math] would be above the horizontal asymptote of [math]g(x)[/math].
  2. If both functions were graphed, the horizontal asymptote of [math]g(x)[/math] would be above the horizontal asymptote of [math]f(x)[/math].
  3. Both functions have the same horizontal asymptote.
  4. Both functions have the same x-intercept.
5. 
The absolute value function [math]f(x)[/math] is graphed below. Comparing [math]f(x)[/math] to [math]g(x) = |3x-1|[/math], which of the following statements is true? For the graph, assume that the scale of both axes is one unit.
Graph - Absolute Function y=|2x|
  1. The minimum value of [math]f(x)[/math] is greater than the minimum of [math]g(x)[/math].
  2. The minimum value of [math]g(x)[/math] is greater than the minimum of [math]f(x)[/math].
  3. The minimum values of the two functions are equal. But if [math]f(x_1)[/math] and [math]g(x_2)[/math] are the minimum values, [math]x_2 > x_1[/math].
  4. The minimum values of the two functions are equal. But if [math]f(x_1)[/math] and [math]g(x_2)[/math] are the minimum values, [math]x_2 < x_1[/math].
6. 
Let [math]f(x) = -1/2 (x-1)(x-3)(x-5)(x-6) [/math]. [math]g(x)[/math] is also a 4th degree polynomial, some of whose approximate values are listed in the table below. Which statement is certainly true?

[math] \ \ \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \ \mathbf{g(x)} \ \ \ \ \ \ \ \ \ \ [/math]
[math] 0 [/math][math] -31.8 [/math]
[math] 1 [/math][math] 1.4 [/math]
[math] 2 [/math][math] 3.3 [/math]
[math] 3 [/math][math] -1.9 [/math]
[math] 4 [/math][math] -2.5 [/math]
[math] 5 [/math][math] 1.8 [/math]
[math] 6 [/math][math] -1.2 [/math]
  1. Both functions have 2 local maximums and 2 local minimums.
  2. Both functions have 4 distinct zeros.
  3. Both functions have the same range.
  4. [math]f(x) >= g(x)[/math] for all values of [math]x[/math].
7. 
Let [math]f(x)[/math] be the square root function in the following graph. Assume that the scale on both axes is one unit. Also, let [math]g(x) = 5*sqrt(-3x - 2) + 2[/math]. Which of the following statements is true about these functions?
Graph - Square Root Function y=sqrt x
  1. The minimum value of [math]f(x)[/math] is less than the minimum value of [math]g(x)[/math].
  2. The minimum value of [math]g(x)[/math] is less than the minimum value of [math]f(x)[/math].
  3. The domain of [math]f(x)[/math] and the domain of [math]g(x)[/math] are identical.
  4. The y-intercepts of both functions are real numbers which are greater than or equal to zero.
8. 
Let [math]f(x)[/math] be the square root function graphed below. Assume that the scale of both axes is one unit. Let [math]g(x) = x^2 - 10x + 24[/math]. On what interval is [math]f(x) > g(x)[/math]. Round answers to the nearest unit, if necessary.
Graph - Square Root Function y=sqrt x
  1. [math](-oo, oo)[/math]
  2. [math][0, 3) uu (7,oo)[/math]
  3. [math](3, 7)[/math]
  4. [math]f(x)[/math] is never greater than [math]g(x)[/math].
9. 
If [math]f(x)[/math] is the function in the graph below, and some of the values of the logarithmic function [math]g(x)[/math] are listed in the table below, approximately for what values of [math]x[/math] is [math]g(x) > f(x) ?[/math] For the graph, assume that the scale of both axes is one unit.

Graph - Exponent Function y=2^x

[math] \ \ \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \ \mathbf{g(x)} \ \ \ \ \ \ \ \ \ \ [/math]
[math] -3.75 [/math][math] -2.7 [/math]
[math] -3.5 [/math][math] -0.9 [/math]
[math] -3.25 [/math][math] -0.3 [/math]
[math] -3 [/math][math] 0.1 [/math]
[math] -2.5 [/math][math] 0.6 [/math]
[math] -2 [/math][math] 0.9 [/math]
[math] -1.5 [/math][math] 1.1 [/math]
[math] -1 [/math][math] 1.3 [/math]
[math] 0 [/math][math] 1.6 [/math]
[math] 1 [/math][math] 1.9 [/math]
[math] 2 [/math][math] 2.1 [/math]
[math] 3 [/math][math] 2.2 [/math]
[math] 4 [/math][math] 2.4 [/math]
  1. [math]g(x)[/math] is never greater than [math]f(x)[/math].
  2. [math]x>1[/math]
  3. [math]x > -4[/math]
  4. [math]-3 < x < 1[/math]
10. 
Let [math]f(x)[/math] be the piecewise function in the graph below. Assume that the scale on both axes is one unit, and that the function is only defined for the values shown in the graph. If [math]g(x) = -x^2 + 7x-6[/math], for what values of [math]x[/math], approximately, is [math]f(x) > g(x) ?[/math]
Graph - Piecewise Random 1
  1. [math]1.5 < x < 5[/math]
  2. [math]-oo < x < 1.5 " or " 5 < x < oo[/math]
  3. [math]0 <= x < 1.5 " or " 5 < x <= 10[/math]
  4. [math]f(x)[/math] is never greater than [math]g(x)[/math].

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