Help with Pre-Calculus?

Question

Sketch the graph of each function and find it's reciprocal: f(x)=x^2+3x+2. 

Answer 1

The function f(x) = x² + 3x + 2

To graph the function, find horizontal and vertical intercepts.

To find x intercepts substitute x = 0 in y = x² + 3x + 2.

y = (0)² + 3(0) + 2 = 2

y intercept is (0, 2)

To find y intercepts substitute y = 0 in y = x² + 3x + 2.

x² + 3x + 2 = 0

x² + 2x + x + 2 = 0

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

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

x = - 2 and x = - 1

x intercepts are (- 2, 0) and (-1, 0).

We need some more points to more accurate graph.

Choose random values for x and find the corresponding values for y.

x	y = x2 + 3x + 2	(x, y)
- 5	y = (-5)2 + 3(-5) + 2 = 12	(- 5, 12)
- 4	y = (-4)2 + 3(-4) + 2 = 6	(-4, 6)
- 3	y = (-3)2 + 3(-3) + 2 = 2	(-3, 2)
1	y = (1)2 + 3(1) + 2 = 6	(1, 6)
2	y = (2)2 + 3(2) + 2 = 12	(2, 12)

Graph :

Draw the coordinate plane.

Plot x, y intercepts and points found in the table.

Connect the plotted points with smooth curve.

.

Answer 2

The function f(x) = x² + 3x + 2

Change f(x) to y.

y = x² + 3x + 2

Interchange x and y.

x = y² + 3y + 2

Solve for y.

The equation y² + 3y + (2 - x) = 0

Compare it to ax² + bx + c = 0.

a = 1, b = 3, c = 2 - x

Roots are x = [- b ± √(b² - 4ac)]/2a

y = [- 3 ± √(3² - 4(1)(2 - x)]/2(1)

y = [- 3 ± √(9 - 4(2 - x)]/2

y = [- 3 ± √(9 - 8 + 4x)]/2

y = [- 3 ± √(1 + 4x)]/2

Change f^-1(x) to y.

f^-1(x) = [- 3 ± √(1 + 4x)]/2.

The original function and it's inverse function are looks like in the coordinate axes.

.