Consider the function fx 2x2 9x 5 Determine the x and y i

Consider the function f(x) = 2x^2 - 9x - 5 Determine the x and y intercepts. Determine the coordinates of the vertex, V(h, x) Make a rough sketch of the graph of f(x). Use the tables below to evaluate each of the following (if possible) g^-1(0) = (fog)(4) =

Solution

7] a]

For y intercept, substitute x = 0 in the function.

f(x) = 2x² - 9x - 5

so f(0) = 2(0)² - 9(0) - 5 = 0 - 0 - 5

therefore the y intercept is c = - 5.

for x intercept, substitute y = 0 in f(x)

therefore  2x² - 9x - 5 = 0

which can be written as:

2x² - 10x + x - 5 = 0

2x[x - 5] + 1[x - 5] = 0

=> [2x + 1][x -5] = 0

therefore x = - 1/2 and x = 5 are the x intercepts.

b]  2x² - 9x - 5 can be written as: 2[x² - (9/2)x - (5/2)] => 2([x - (9/4)]² - 7.5625)

so the vertex of this function will be at x = 9/4.

c] The graph is an upward parabola with its lowest point at x = 9/4, its y intercept with a value c = - 5 and x intercepts at x = -1/2 and x = 5.

8] from the table, g(-1) = 0

therefore g^-1 (0) = - 1

f o g (4) is the same as f(g(4))

from the table, g(4) = 5

and so f(g(4)) = f(5)

from the above table, f(5) = not defined since the table ends at x = 4.

therefore f o g (4) cannot be detemined.

g o f^-1 (5) = g[f^-1(5)]

from the above table, f(2) = 5

therefore f^-1(5) = 2

so g[f^-1(5)] = g[2]

from the table, g[2] = 1

therefore g o f^-1 (5) = 1