A function is always recognized by
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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.

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:

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.

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.

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.

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.`