ACT Math: true relationship between f(x) = (x - 3)^2 + 2 and g(x) = (1/2)x + 1

[mjmax333]

Hi! I'm studying to take the ACT a third time! Here's a question from a practice test I have no idea how to do:

Can anyone explain how to do this problem, or provide a resource teaching me how to do this type of problem? I don't necessarily just want someone to tell me the answer, I need to know how to do the problem! Thank you so much

 

[ksdhart2]

Okay, well, the image is a bit small, but here's what I think it says. If I'm wrong please reply with any necessary corrections.

Which of the following describes a true relationship between the functions \(\displaystyle f(x)=(x-3)^2+2\) and \(\displaystyle g(x)=\dfrac{1}{2}x+1\), graphed below in the standard (x,y) coordinate plane?

F. f(x)=g(x) for exactly 2 values of x
G. f(x)=g(x) for exactly 1 value of x
H. f(x) < g(x) for all x
J. f(x) > g(x) for all x
K. f(x) is the inverse of g(x)

Since this is a multiple choice question and it asks you to determine which is/are (a) true statement(s), I think the best way is to evaluate each statement and see if it's true or not. Let's start with F and G. What does it mean for two functions to be equal? Are there any values of x for which the two functions are equal? If so, how many such values are there? Are either of F or G true? Similarly for H and J, what does it mean for one function to be greater than or less than another? Is one of the functions always less than another? If so, which function is smaller? What does that suggest about whether H or J is true? Finally, for K, how do you know if a function is invertible or not? If f(x) is the inverse of g(x), then it follows that g(x) is the inverse of f(x). So, are both f(x) and g(x) invertible? If so, are they both each other's inverse?

 

[Steven G]

Hi! I'm studying to take the ACT a third time! Here's a question from a practice test I have no idea how to do:

Can anyone explain how to do this problem, or provide a resource teaching me how to do this type of problem? I don't necessarily just want someone to tell me the answer, I need to know how to do the problem! Thank you so much

Graphing the 2 functions will help and give the correct answer.

If you can't graph the two functions then:

f(0)=(0-3)² + 2 = (-3)² + 2 = 9 + 2 =11
\(\displaystyle g(0)=\dfrac{1}{2}*0 + 1\)
so f(0)>g(0)

f(3)=(3-3)² + 2 = (-0)² + 2 = 0 + 2 =2
\(\displaystyle g(3)=\dfrac{1}{2}*3 + 1 =\dfrac{5}{2}\)
so f(3)<g(3)

This rules out h and j

f(5)=(5-3)² + 2 = (2)² + 2 = 4 + 2 =6
\(\displaystyle g(5)=\dfrac{1}{2}*5 + 1 =\dfrac{7}{2}\)
so f(5)>g(5)
The only way f and g can change which is larger twice is if there are two points of intersection. So Choice B is the answer.

If you know how to graph quadratic equation and lines then let us know where you had trouble.

You can always set the eqs equal to one another and solve but if you can do that then you should be able to graph the eqs.

One last thing. I randomly picked x to be 0, 3 and 5 above. It might take a few tries getting the right 3 x-values to show that the graphs intersection two places.

 

[pka]

Hi! I'm studying to take the ACT a third time! Here's a question from a practice test I have no idea how to do:

Can anyone explain how to do this problem

You ask how to do this problem, the answer is simple: look carefully at the graph.
This question is testing your knowledge of linear graphs & parabolic graphs.
Lines have two sides in a plane. If the other function intersects the line it cannot be all greater than or all less than the line. The inverse is a line is a line. Now those take care of H, J, K. So which distractor left is correct?

 

[mjmax333]

Graphing the 2 functions will help and give the correct answer.

If you can't graph the two functions then:

f(0)=(0-3)² + 2 = (-3)² + 2 = 9 + 2 =11
\(\displaystyle g(0)=\dfrac{1}{2}*0 + 1\)
so f(0)>g(0)

f(3)=(3-3)² + 2 = (-0)² + 2 = 0 + 2 =2
\(\displaystyle g(3)=\dfrac{1}{2}*3 + 1 =\dfrac{5}{2}\)
so f(3)<g(3)

This rules out h and j

f(5)=(5-3)² + 2 = (2)² + 2 = 4 + 2 =6
\(\displaystyle g(5)=\dfrac{1}{2}*5 + 1 =\dfrac{7}{2}\)
so f(5)>g(5)
The only way f and g can change which is larger twice is if there are two points of intersection. So Choice B is the answer.

If you know how to graph quadratic equation and lines then let us know where you had trouble.

You can always set the eqs equal to one another and solve but if you can do that then you should be able to graph the eqs.

One last thing. I randomly picked x to be 0, 3 and 5 above. It might take a few tries getting the right 3 x-values to show that the graphs intersection two places.

Hi, I still don't understand how you were able to decide that the graph intersects at 2 places.

 

[mjmax333]

You ask how to do this problem, the answer is simple: look carefully at the graph.
This question is testing your knowledge of linear graphs & parabolic graphs.
Lines have two sides in a plane. If the other function intersects the line it cannot be all greater than or all less than the line. The inverse is a line is a line. Now those take care of H, J, K. So which distractor left is correct?

Where does it say that the function intersects? I'm so confused.

 

[ksdhart2]

Where does it say that the function intersects? I'm so confused.

The problem doesn't explicitly state, in words, that the two functions intersect, but you can easily glean that information simply by looking at the graph provided. Or, another way to tell is to understand what it means for two functions to intersect - it means that they are equal for some value(s) of x. So, if you set the two functions equal and solve for x, you'll find the point(s) of intersection. I generally prefer the algebraic method, as just looking at the graph is usually only sufficient to give an estimate of where the functions intersect, but solving for x spits out the exact values.

 

[mjmax333]

The problem doesn't explicitly state, in words, that the two functions intersect, but you can easily glean that information simply by looking at the graph provided. Or, another way to tell is to understand what it means for two functions to intersect - it means that they are equal for some value(s) of x. So, if you set the two functions equal and solve for x, you'll find the point(s) of intersection. I generally prefer the algebraic method, as just looking at the graph is usually only sufficient to give an estimate of where the functions intersect, but solving for x spits out the exact values.

How do you solve for x for these kind of problems? I'm trying and I can't do it.

 

[Deleted member 4993]

How do you solve for x for these kind of problems? I'm trying and I can't do it.

[h=2]f(x) = (x - 3)^2 + 2 and g(x) = (1/2)x + 1[/h](x-3)^2+2=x/2+1

x^2 - (13)/2 * x + 1 = 0

Above is a quadratic equation. Solve it using your favorite method (quadratic formula, completing square, etc.).

 

[stapel]

Where does it say that the function intersects? I'm so confused.

It doesn't. You're expected to know this. From the graphing and solving you've done, you were expected to have learned that points of intersection are points where two lines are in the exact same place. In this case, they are points where f(x) and g(x) are in the exact same place. If they're in the exact same place, then they must have the same values, so they must be equal, so it must be true that f(x) = g(x).

 

[mjmax333]

It doesn't. You're expected to know this. From the graphing and solving you've done, you were expected to have learned that points of intersection are points where two lines are in the exact same place. In this case, they are points where f(x) and g(x) are in the exact same place. If they're in the exact same place, then they must have the same values, so they must be equal, so it must be true that f(x) = g(x).

W H A T
I am no less confused.

 

[mjmax333]

f(x) = (x - 3)^2 + 2 and g(x) = (1/2)x + 1

(x-3)^2+2=x/2+1

x^2 - (13)/2 * x + 1 = 0

Above is a quadratic equation. Solve it using your favorite method (quadratic formula, completing square, etc.).

Okay, I know how to do the quadratic formula, but how do I know which numbers from the above equations to submit for the variables of the quadratic fromula? I.e. How do I find A, B, and C.

 

[ksdhart2]

Okay, I know how to do the quadratic formula, but how do I know which numbers from the above equations to submit for the variables of the quadratic fromula? I.e. How do I find A, B, and C.

The variables a, b, and c are the coefficients of the quadratic equation you're attempting to solve. The generic form is: ax² + bx + c. a is the coefficient of (the number being multiplied by) x², b is the coefficient of x, and c is the last, free-standing term. Now, your specific quadratic equation is x² + 13/2 * x + 1. So which number is being multiplied by x²? Which number is being multiplied by x? What's the free-standing term? Those are your a, b, and c.

 

[mjmax333]

The variables a, b, and c are the coefficients of the quadratic equation you're attempting to solve. The generic form is: ax² + bx + c. a is the coefficient of (the number being multiplied by) x², b is the coefficient of x, and c is the last, free-standing term. Now, your specific quadratic equation is x² + 13/2 * x + 1. So which number is being multiplied by x²? Which number is being multiplied by x? What's the free-standing term? Those are your a, b, and c.

Okay, I got -.158 and -6.343. I feel like this is wrong. What did I do wrong??

 

[ksdhart2]

Any particular reason you think those answers aren't right? Is it because both of them are negative? If so, that's absolutely allowed. Sometimes polynomials have only negative roots, sometimes they have only positive roots (and sometimes they just don't have any [real] roots). In the future, if you're ever unsure of your answer, you can check it yourself. You know that x² + (13/2)x + 1 = 0. Your proposed solutions are approximately -0.158 and -6.343, so let's plug those in and see what happens:

(-0.158)² + (13/2)(-0.158) + 1 = -0.002036
(-6.343)² + (13/2)(-6.343) + 1 = 0.004149

Those values are very close to 0, so these are probably the roots of the polynomial, but we can't know for sure. This is the problem with using rounded or approximate answers - best to stick with them in their original form. In the case of the quadratic formula, you'll end up with [(something) +/- (square root of something)]/(something). If you plug those values into the polynomial, do you get exactly 0? If not, you'll have to share your work with us so we can point out where you may or may not have gone wrong.

 

[mmm4444bot]

x^2 - (13)/2 * x + 1 = 0

That equation should be: x^2 - (13/2)x + 10 = 0

 

[stapel]

W H A T
I am no less confused.

Okay. So is it "graphing" that you've not done yet? Or "solving" equations? Or you're not familiar with what "points" are? Or what "the same place" means?

We'll be glad to try to clarify what's going on, but we'll need to know which parts, specifically, are not familiar to you. Only then can we begin to try to fill the gaps in your background knowledge.

Thank you!