How man xintercepts can the graph of a function have How man

How man x-intercepts can the graph of a function have? How many y-intercepts can the graph of the function have? Explain in full sentences. Find the vertical, horizontal, and oblique (slant) asymptotes, if any, of the rational function R(x) = 6x^2 + x + 12/3x^2 - 5x - 2 Given the function h(x) = squareroot x, perform the transformations (a), (b), and (c) below and graph the final result (make sure to write your resulting transformation in every step). Use the grid provided in Figure 3 for your convenience. Shift the original function up 1 unit. Write down your expression for the new function. Reflect the result of part a) about the x-axis. Write down your expression for the new function. Reflect the result of part b) about the y-axis. Write down your final expression.

Solution

1.a. A graph can have many x intercepts and y intercepts. But to be a function,it must pass the vertical line test that is if the vertical line hits more than once,then it is not a function. Therefore in case of function it must have only one y intercept.

b. f(x) =   (6x²+x +12)/(3x²-5x-2) To find the vertical asymptote,we have to set the denominator to zero and solve for x

3x²-5x-2=0

(3x+1)(x-2)=0

x=-1/3 and x=2

Therefore the vertical asymptotes are x=-1/3 , x=2

In the given expression the degree of numerator and denominator both are 2 that is they are same

In that case horizontal asymptote, y= numerator\'s leading coefficient/denominator\'s leading coefficient

                                                     y=6/3=2

Therefore the horizontal asymptote is y=2

There is no slant asymptote here