How do you graph # f(x)= x/(x+3)#?

Answer 1

graph{1-3/(x+3) [-10, 10, -15, 15]}

First of all, we see that the function is not defined at #x=-3#. In fact, there is a vertical asymptote at this point since the denominator tends to #0# as #x->-3#, while the numerator is around a constant #-3#.
More than that, as #x->-3# from the right (that is, while it's greater than #-3#), the sign of the function is negative since the numerator is around #-3# and denominator is positive, so the function tends to negative infinity #-oo#. If #x->-3# from the left (that is, while it's smaller than #-3#), the sign of the function is positive since the numerator is around #-3# and denominator is negative, so the function tends to positive infinity #+oo#.
Continuing analysis, we can see that, as #x->+oo# or as #x->-oo#, the value of the function tends to #1# since both numerator and denominator will increase to #+oo# or decrease to #-oo# with the same speed.
Now about constructing a graph. The easiest way is to transform our function as follows: #y=x/(x+3)=(x+3-3)/(x+3)=(x+3)/(x+3)-3/(x+3)=1-3/(x+3)#
So, we have to graph #y=1-3/(x+3)#. According to the rules of graph transformation (seeUnizor - Algebra - Graph) we can construct this graph in the following steps:
Step 1. Graph #y=1/x#

graph{1/x [-10, 10, -15, 15]}

Step 2. Shift it to the left by #3# to graph #y=1/(x+3)#

graph{1/(x+3) [-10, 10, -15, 15]}

Step 3. Stretch it vertically by a factor of #3# getting the graph of #y=3/(x+3)#

graph{3/(x+3) [-10, 10, -15, 15]}

Step 4. Invert the graph (positive - to negative, negative - to positive), thus getting the graph of #y=-3/(x+3)#

graph{-3/(x+3) [-10, 10, -15, 15]}

Step 5. Finally, shift the graph up by #1# to get #y=1-3/(x+3)#:

graph{1-3/(x+3) [-10, 10, -15, 15]}

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Answer 2

To graph the function f(x) = x/(x+3), follow these steps:

  1. Determine the domain of the function, which is all real numbers except for x = -3 (since division by zero is undefined).

  2. Find the y-intercept by substituting x = 0 into the equation: f(0) = 0/(0+3) = 0/3 = 0. So, the y-intercept is (0, 0).

  3. Determine the x-intercept by setting f(x) = 0 and solving for x: x/(x+3) = 0. This occurs when x = 0, so the x-intercept is (0, 0).

  4. Analyze the behavior of the function as x approaches positive and negative infinity. As x approaches infinity, f(x) approaches 1. As x approaches negative infinity, f(x) approaches -1.

  5. Plot additional points by choosing various x-values and calculating the corresponding y-values using the equation.

  6. Draw a smooth curve passing through the plotted points, considering the behavior of the function and the asymptotes.

  7. Finally, label the x and y axes, and any other relevant points or features on the graph.

Note: The graph will have a vertical asymptote at x = -3, as the function approaches infinity as x approaches -3 from the left, and approaches negative infinity as x approaches -3 from the right.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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