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Graphing Functions


A function is always recognized by its graph but they are not same because it happens that two Functions that have different value of co- domain must have same graph. If we test graph of a function then we use vertical line test and if we test whether function is one to one then we use horizontal line test. If we have Inverse Function then graph of inverse function can be obtained by plotting graph of function along the line q = p. (here 'p' is along x - axis and 'q' is along y - axis. In a graph a curve is one to one function if and only if it is a function. For example: Suppose we have a function:
F (x) = p, if p = 1,
F(x) = q, if p = 2,
F(x) = r, if p = 3. Then Graphing Functions of given values.

Here we get function value (1, p), (2, q), (3, r). We can also write function in cubical form as:

F (p) = $p^3$ - 9p, on plotting graph of this function, we get curve shown below:

 Graphing Function
This is how we plot functions on a graph.

Linear  functions

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A linear function is of the form ax + by + c = 0. The graph of a linear function is a straight line. To plot the graph the given steps should be taken:

1) Find x-intercept and y-intercept.

2) Find at least one random point that satisfies the equation.

3) Plot the intercepts and the point obtained.

4) Join the three points to get the straight line.

To get x-intercept, find the point where the line intercepts the x-axis by putting y = 0. Similarly, for y-intercept put x = 0. For getting the random point, take a random point for x, and put in the equation to get the corresponding value of y. Given is the graph for the linear function 3x + 2y + 1 = 0.

Linear Function

Logarithmic functions

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To graph a logarithmic function, we first need to understand the function itself. The logarithm function is the inverse of the exponential function. We have, $2^{3} = 8$.
From there we can get $3 = log_{2}8$. To graph a logarithmic function the given points should be taken care of:

1) Find the asymptote of the function.

2) Find the domain of the function.

3) Find the x-intercept of the function.

Suppose we have been given a function $y=log_{2}(x+1)$. The domain of a f(x) = log x function is [0, $\infty$]. For $y=log_{2}(x+1)$ the domain will be x + 1 = 0 to x + 1 = $\infty$. Hence, domain is [-1, $\infty$].

For the log x function the asymptote is x = 0. For $y=log_{2}(x+1)$ the asymptote is x = -1.

For getting the x-intercept, $ 0 = log_{2}(x + 1)$ which gives $2^{0} = x + 1$ which gives x = 0. The graph will be like given below:

Logarithmic Function

Exponential functions

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Exponential function grows exponentially as the name suggests. The domain is of all real numbers and the range is all positive real numbers. The asymptote to an exponential function, $y= a^{x}$ is x = 0, that is, the Y-axis.

To graph the functions follow the given steps:

1)  Find intercepts.

2)  Get random points of x, and obtain corresponding y-coordinates.

3) Plot the graph on those points.

For Example: Take a function $y=3^{x}$. Putting x = 0, y intercept comes as 1. The graph will be like as given here.

Exponential Function

Inverse functions

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The graph of the inverse of a function is symmetrical with the function with respect to the line y = x. The inverse of a function will be having same points as the function itself but the values of x and y will be reversed. For example: if the function has a point (1, 2) on it the inverse will have a point (2, 1) on it. For a function: $f(x) = 2x + 2$, the inverse will be $g(x) = \frac{x-2}{2}$. For f(x), if x = 2, f(x) = 6. hence, the point is (2, 6) in the Cartesian plane. Putting, x = 6 in g(x) it becomes 2. The point becomes (6, 2).
We can see that the coordinates are being exchanged:
To graph the inverse of a function, make the graph of the function reflect about the line y = x.The graph for the example mentioned above will be as given here. We can see that the two graphs are mirror images with respect to the line y = x.

Inverse Function

We can see that all three graphs are intersecting at the point (-2, -2). Can you reason why? Because this point will satisfy all the three equations.

Rational Functions

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A rational function is one having a fraction in it. For example: f(x) = $\frac{p(x)}{q(x)}$. The graph of a rational function will never cross X-axis.

For getting the graph of a rational function following points should be followed:

1) Find x-intercept by putting numerator equal to zero.

2) Find y-intercept by putting denominator equal to zero.

3) Find vertical asymptote by putting denominator equal to zero.

4) Find horizontal asymptote by dividing leading coefficients.

For Example: For the equation $y = $\frac{x - 3}{x+1}$ the graph will be as given here.`

Rational Function