Havar D.

asked • 11/21/22

Solving functions out from a graph - finding the image of a function - curve intervals

The question states: "The image below shows the curve of the function f(x) where Df = [-2,2]. Use the graph to answer the following questions: (I have linked said image)


a) find the image of function f


b) solve the equation f(x) = 0


(Here I would assume y = 0, given the graph)


c) solve the equation f'(x) = 0


d) solve the equation f''(x) = 0


e) at what interval does the function curve downward?


Here is the graph attached:


ibb.c o/yXTX407 (please remove the spaces between the c and o, otherwise it will not let me upload)


Again any help would be appreciated.


Mark M.

What image below? Provide all information if you want some help.
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11/21/22

Havar D.

ibb.c o/yXTX407 (please remove the spaces between the c and o, otherwise it will not let me upload)
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11/21/22

Havar D.

Did it work?
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11/21/22

Daniel B.

tutor
The function appears to be x^2(x^2 - 3). But it does not need to be exactly. Would you know how to proceed if knew the expression for f(x). Do you need help in figure out the expression for f(x).
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11/24/22

David L.

tutor
This belongs under calculus, not precalculus.
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05/14/23

Brenda D.

tutor
It’s hard to see the details of the numbers, even in an expansion of the image but there are some features that should help you get started. You have a graph that is symmetrical about the origin, has two minima, passes through the origin, with two other x intercepts that may be square roots of some number greater than 1 but less than 4. Your x intercepts look close to +/- 1.7 or +/- 1.8 but not +/-2 and you still have x=0. So you probably have x^3 up to x^4 per the number of roots. Also since your graph passes through (0,0) it quite possibly represents the difference between two squares, where there is no middle term as opposed to something like x^4 -5x^2 +4 that would have 4 visible x intercepts. Except for passing through the origin, nothing else that could be considered a y intercept. While I could not see the numbers very well, I saw enough to compare it to x^4 - 4x^2 this would look exactly like your graph with x intercepts at (-2,0),(0,0)(2,0), minima at (-1.414, -4), (1.414, -4). If you can see x intercepts and minima values better on your paper; these are numbers you can try yourself against the Quadratic Formula or Completing the Square. Check to see if the values remind you of any square roots that you know or can look up. Last but not least did you compare your graph to those of some familiar polynomials like a Compound Quadratic. Sorry for the long note.
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07/08/23

1 Expert Answer

By:

Jonathan T. answered • 10/26/23

Tutor
5.0 (362)

Calculus, Linear Algebra, and Differential Equations for College

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a) To find the image of the function f, you need to determine the range of values that the function f(x) takes on. Examine the vertical extent of the graph, which represents the range of f(x). Identify the lowest and highest points on the graph to determine the range.


b) To solve the equation f(x) = 0, you're looking for the x-values where the function crosses the x-axis on the graph. These are the points where f(x) is equal to zero.


c) To solve the equation f'(x) = 0, you are looking for the critical points of the function. These are the x-values where the derivative of the function, f'(x), is equal to zero. Critical points indicate potential maxima, minima, or points of inflection on the graph.


d) To solve the equation f''(x) = 0, you are looking for inflection points. These are the x-values where the second derivative of the function, f''(x), is equal to zero. Inflection points are where the concavity of the graph changes.


e) To determine at what interval the function curve is downward, look for intervals where the slope (derivative) of the function is negative. This corresponds to parts of the graph where the function is decreasing.


If you can provide additional information or describe the graph in more detail, I can offer more specific guidance. If possible, you may want to describe the key features of the graph, such as where it crosses the x-axis, any maximum or minimum points, and where the curvature changes.

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