Relations and Functions - mathematical concepts

The relation is g(x) = x+2 where the 2 is an integer 

This a graph of variable x and added to a constant 2. The graph is a straight line passing though (-2, 0) and (0, 2). The straight line has drifted by +2 from the origin (0, 0). The majority of the first and third quadrant with only a tiny bit crossing through the second quadrant. 

The highest point on the graph occur at the vertex (10, 12) and the graph is symmetrical across y = -x. The y intercept is at the point ( -2, 0) and the x intercept is the point (2, 0). The domain and the range are all real numbers. 

The relation is a function because ii passes the vertical line test, and each domain and range , it has one-to-one correspondence. We can also see that in our function we have dependent part (x+2) and the independent part (g(x)) which is particularly called y. therefore y = g(x). x is the range and y is the domain. 

Transformation 

We shift the units in the function g(x) = x+2 three times upwards and 4 units on the left side. 

Here is how the equation will be affected, 

Three units upwards implies that we add a positive 3 outside the variable x. 

Four units to the left means we add a positive 4 inside of the absolute value bars 

Thus, the function will be g(x) = (x+4) + 3+2 = x + 9 

Basically, adding a constant (say c) to the x-value will move the function to the left (negative direction) and adding a constant (say c) to the existing constant (in this case we have 2) will move the function upwards if and only if c > 0. If C < 0, the function will move downwards. This does incorporate the two shifts into the transformed function. 

#38 

The relation is where the -4 is included with x under the radical. 

After plotting, we a get a graph of a constant 4 and a square root function . Its shaped like a curve which is nearing to be a straight line. it has a starting point at vertex (4, 4) and extend infinitely in the positive infinity direction (to the right). The graph lies entirely in the first quadrant and does not exist in any other three quadrants. 

The graph has neither y intercept nor x intercept. The reason for choosing my set of inputs is because I needed the output to be integers. The domain in this case is all real numbers greater than or equal to 4. This because of the radical ( part. The range is all non-negative real numbers starting from 4 because of the constant 4 in the graph. In the interval notation, we have: 

This relation is a function because every element of the domain has one to one correspondence with the with every element in the range . It also passes the vertical line test ( Abramowitz and Stegun, 1966) . 

If we are asked to transform the function three units up and 4 units to the left we will have 

Work Cited 

Abramowitz, M., & Stegun, I. A. (1966). Handbook of mathematical functions.    Applied mathematics series ,    55 (62), 39.