f(x)= 1/(x+2)
i have to graph this, can some one please help me!
f(x)={ x+5, if x>0
3, if x<=0
x , f(x)
-5 , -1/3
-4 , -1/2
-3 , -1
-2.5 , - 2
-2 undefined denominator of zero
-1.5 , +2
-1 , +1
0 , +1/2
1 , +1/3
Of course! I can help you graph the function f(x) = 1/(x+2). Let's break down the process step by step:
Step 1: Determine the domain of the function.
The domain represents all the possible x-values for which the function is defined. In this case, since we have a fraction and the denominator cannot be zero, the function is undefined when x = -2. Therefore, the domain for this function is all real numbers except x = -2.
Step 2: Determine the vertical asymptote.
The vertical asymptote is a vertical line that the graph approaches but never touches. For this function, the vertical asymptote occurs at x = -2 because the denominator becomes zero when x = -2. So, draw a vertical dotted line at x = -2.
Step 3: Determine the horizontal asymptote.
The horizontal asymptote is a horizontal line that the graph approaches as x approaches positive or negative infinity. To find the horizontal asymptote, we need to examine the behavior of the function as x approaches positive and negative infinity. As x becomes larger and larger, the fraction 1/(x+2) becomes closer and closer to zero. Therefore, the horizontal asymptote is y = 0 or the x-axis.
Step 4: Plot the x- and y-intercepts.
To find the x-intercept, we set y = 0 and solve for x:
0 = 1/(x+2)
Solving this equation gives x = -2. So, the x-intercept is (-2, 0).
To find the y-intercept, we set x = 0 and evaluate the function:
f(0) = 1/(0+2) = 1/2
So, the y-intercept is (0, 1/2).
Step 5: Plot additional points and sketch the graph.
Now that we have the key points (the vertical asymptote, horizontal asymptote, x-intercept, and y-intercept), you can plot them on the graph. In addition, you can choose a few other x-values, calculate the corresponding y-values using the function, and plot those points as well. For example, you can try x = -3, -1, and 1.
After plotting these points, connect them smoothly to form the graph. Keep in mind that as x gets closer to -2 (but not equal to -2), the function becomes very large (positive or negative). This behavior is reflected in the graph.
I hope this step-by-step explanation helps you graph the function f(x) = 1/(x+2) successfully.