Understanding the Intersection of Functions (Grades 11-12)

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Understanding the Intersection of Functions

1. 
Given the functions [math]f(x) = 3x+2[/math] and [math]g(x) = 5x-2[/math], for which value of [math]x[/math] is [math]f(x)=g(x) ?[/math]
  1. [math]8[/math]
  2. [math]2[/math]
  3. [math]-2[/math]
  4. [math]1/2[/math]
2. 
The graphs of the functions [math]f(x)=|x-2|[/math] and [math]g(x) = 1/2 x+1[/math] intersect at two points, [math](2/3,4/3)[/math] and [math](6,4)[/math]. Using this information, what is the solution to the equation [math]|x+3| = x-1 ?[/math]
  1. [math]x = 4/3, 4[/math]
  2. [math]x = 6[/math]
  3. [math]x=-3, 2[/math]
  4. [math]x = 2/3,6[/math]
3. 
What value of [math]x[/math] results in equal outputs for the following functions? Choose all correct values.
[math]f(x) = 2|x-4|+1[/math]
[math]g(x)=1/3 x + 2 [/math]
  1. [math]x=3[/math]
  2. [math]x=27/5[/math]
  3. [math]x=-15/7[/math]
  4. [math]x=6[/math]
4. 
The functions [math]f(x) = 2x^3-6x^2-(23/2)x+18[/math] and [math]g(x) = 1/2 x + 2[/math] are graphed on the same axes. Which of the following are true statements about the system of equations [math]y=f(x), y=g(x) ?[/math] There may be multiple correct answers.
  1. [math](1, 5/2)[/math] is a solution since it lies on the graph of both functions.
  2. The points [math](-1, 43/2), (0, 18),[/math] and [math](2,3)[/math] are solutions because they lie on the graph of one or the other function.
  3. [math](3,7/2)[/math] is a solution since [math]g(3)=7/2[/math].
  4. The points [math](-2,1)[/math] and [math](4,4)[/math] are solutions since [math]f(-2)=g(-2)[/math] and [math]f(4)=g(4)[/math].
5. 
The functions [math]f(x)=3x^2-5x-2[/math] and [math]g(x)=2x^2-2x+8[/math] are graphed on the same set of axes. What is true about the points of intersection of the two functions? Choose all correct answers.
  1. Without seeing the graph, there is no way to determine if there are any intersection points.
  2. The intersection points occur where [math]f(x)=g(x)[/math].
  3. [math]f(5)=48[/math] and [math]g(5)=48[/math], so 5 is an x-coordinate of a point where the two functions intersect.
  4. [math]f(0)=-2[/math] and [math]g(-2)=20[/math], so [math](0,20)[/math] is a point of intersection.
6. 
If the graphs of two functions, [math]f(x)[/math] and [math]g(x)[/math], intersect at point [math](a,b)[/math], what is the value of [math]f(a)-g(a)?[/math]
  1. [math]b[/math]
  2. [math]0[/math]
  3. [math]f(b)[/math]
  4. [math]g(b)[/math]
7. 
You graph two functions, [math]f(x)[/math] and [math]g(x)[/math], and they intersect in three places. Which of the following statements MUST be correct?
  1. Neither function is linear.
  2. Solving [math]f(x)=x[/math] or [math]g(x)=x[/math] will give the x-coordinate of the points of intersection.
  3. There are three distinct x values for which [math]f(x)=g(x).[/math]
  4. There are three distinct y values, or output values, for which [math]f(x)=g(x) .[/math]
8. 
The functions [math]f(x) = e^x+1[/math] and [math]g(x) = (x^2-3)/(x+2)[/math] are graphed on the same set of axes. To find where the functions intersect, the following table of values was calculated for the functions.

[math] \ \ \ \ \ \ \ \ \mathbf{x} \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \mathbf{g(x)} \ \ \ \ \ \ \ [/math]
[math]-1.9[/math][math]1.1496[/math][math]6.100[/math]
[math]-1.8[/math][math]1.1653[/math][math]1.2000[/math]
[math]-1.7[/math][math]1.1827[/math][math]-0.3667[/math]


Which of the following statements about the point of intersection of the two functions is correct? Choose all correct answers.
  1. The x-coordinate of the point of intersection lies between 1.1 and 1.2.
  2. One would need x values of greater precision, such as -1.85 or -1.775, to calculate a more precise answer.
  3. The x values chosen are not near the point of intersection, so we cannot determine the point of intersection.
  4. The x-coordinate of the intersection point is approximately -1.8.
9. 
Jessica graphs the functions [math]f(x)=3|x-4|[/math] and [math]g(x)=3log_{10}(5x)[/math] to find the points of intersection. She sees that one point of intersection occurs between [math]x=2.5[/math] and [math]x=3[/math]. She also notices that, in this interval, [math]f(x)[/math] is decreasing and [math]g(x)[/math] is increasing. Next, she creates the following table of values.

[math] \ \ \ \ \ \ \ \ \mathbf{x} \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \mathbf{g(x)} \ \ \ \ \ \ \ [/math]
[math]2.6[/math][math]4.200[/math][math]3.342[/math]
[math]2.7[/math][math]3.900[/math][math]3.391[/math]
[math]2.8[/math][math]3.600[/math][math]3.438[/math]
[math]2.9[/math][math]3.300[/math][math]3.484[/math]


What can she determine from this table of values?
  1. More information is needed to make any conclusion.
  2. The intersection point lies on [math]2.6 < x < 2.7[/math].
  3. The intersection point lies on [math]2.7 < x < 2.8[/math].
  4. The intersection point lies on [math]2.8 < x < 2.9[/math].
10. 
Simone sketches the functions [math]f(x) = x^3 + 4x^2 + 2x - 9[/math] and [math]g(x) = e^(x-5) - 7.[/math] She sees that one of the intersection points occurs between [math]x=-4[/math] and [math]x=-2.[/math] She also notices that, near this intersection point, to the left of the intersection point, [math]f(x)[/math] is below [math]g(x)[/math] on the graph, and to the right of the intersection point [math]f(x)[/math] is above [math]g(x)[/math].
A. 
In order to get a more precise location of the point of intersection, she decides to try the value of [math]x[/math] in the middle of the interval. What are the values of each function at this point, rounded to two decimal places?
  1. [math]f(-3) = -6.00, \ \ g(-3)=-7.00[/math]
  2. [math]f(-3) = -78.00, \ \ g(-3)=-7.00[/math]
  3. [math]f(-3) = -78.00, \ \ g(-3) = -0.39[/math]
  4. [math]f(-3)= 0.00, \ \ g(-3) = -6.86[/math]
B. 
Given the values of the functions at [math]x=-3[/math], and the relative values of [math]f(x)[/math] and [math]g(x)[/math] near this point, should Simone choose a value greater than or less than [math]x=-3[/math] to get a more precise value of the point of intersection?
  1. Neither, [math]x=-3[/math] is the point of intersection.
  2. Less than, since [math]f(-3) > g(-3)[/math].
  3. Greater than, since [math]f(-3) > g(-3) [/math].
  4. Either, since it is still just an approximation.
C. 
If Simone continues to choose values of [math]x[/math] in the middle of each new interval (for example, she would choose [math]x=-2.5[/math] if she looks in the interval [math]-3 < x < -2[/math]), what would be a reasonable criterion for believing that she has found an accurate point of intersection and why?
  1. [math]|f(x)|[/math] is very small, because the [math]x[/math] value is very precise.
  2. [math]|g(x)|[/math] is very small, since [math]x[/math] is near the middle of the interval.
  3. [math](|f(x)+g(x)|)/2[/math] is near zero, as you've averaged the absolute value of the functions and the functions should cancel each other out.
  4. [math]|f(x)-g(x)|[/math] is very small, since these values should be almost equal near the point of intersection.

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