use the function f(x)=x^2+4x+1, h(x)= 11/x^2-2x

Question

Find h(3)
Find the x and y intercepts for f(x)
Find domain of h(x)

Answer 1

h(x) = 11 / (x² - 2x)

 

Substitute x = 3

 

h(3) = 11 / (3² - 2(3))

h(3) = 11 / (9 - 6)

       = 11 / 3

 

 

To find x-intercepts of f(x), set f(x) = 0.  To find y-intercepts of f(x), evaluate f(0).

 

0 = x² + 4x + 1

 

Use the quadratic formula to solve for x. 

 

x = (-4 ± √(16 - 4)) / 2

x = (-4 ± √(12)) / 2

x = (-4 ± 3.46) / 2

 

x = (-4 + 3.46) / 2        and        x = (-4 - 3.46) / 2   

 

x = -0.27                     and         x = -3.73

 

x-intercepts are  (-3.73, 0) and (-0.27, 0).

 

f(0) = 0²  4(0) + 1

       = 1

 

y-intercept is (0, 1).

 

 

To find the domain of h(x), set the denominator of h(x) equal to zero.

 

x² - 2x = 0

x(x - 2) = 0

 

x = 0      and     x = 2

 

At these x values, the function is undefined.

 

Domain is    (-∞, 0)∪(0, 2)∪(2, ∞).

Answer 2

h(3) means we plug 3 into the equation:

h(3) = 11 / (3² - 2*3)

h(3) = 11 / (9 - 6)

h(3) = 11 / 3

 

To find the y-intercept, we start off by setting f(x) = 0

f(0) = 0 + 0 + 1

f(0) = 1

 

To find the x-intercepts, we need to factor first. Unfortunately, there's no easy way to get M * N = 1 and M + N = 4, so we need to use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

x = (-4 ± √(4² - 4 * 1 * 1)) / 2 * 1

x = (-4 ± √(16 - 4)) / 2

x = (-4 ± √8) / 2

x = (-4 ± √4*2) / 2

x = (-4 ± 2√2) / 2

x = -2 ± √2

And those are our two x-intercepts

 

To find the domain of h(x), we need to find at what points does the denominator of h(x) = 0, because those will be the points that will not be part of the domain:

x² - 2x = 0

x(x - 2) = 0

So when x = 0 or 2, we will have a 0 in the denominator and will end up dividing by 0.

So the domain is all x not equal to 0 or 2.