Question

F(x)=x^2-4
g(x)=2x+1

solve fg(x)>0
SHOW WORKING

Answer 1

To solve the inequality fg(x) > 0, find values of x such that f(x) = 0 and determine intervals where fg(x) > 0.

Step-by-step explanation:

To solve the inequality fg(x) > 0, we need to find the values of x for which the product of f(x) and g(x) is greater than zero.

First, we find the values of x where f(x) = 0, by solving the equation x^2 - 4 = 0. We factor it as (x - 2)(x + 2) = 0, and we get x = -2 and x = 2.

Next, we evaluate the sign of f(x) and g(x) in the intervals defined by these values of x. We plug in a value less than -2, between -2 and 2, and greater than 2 into both f(x) and g(x). From this, we can determine the intervals where fg(x) > 0. The solution is (-∞, -2) ∪ (2, ∞).

Answer 2

To find the values of x that make fg(x) > 0, we first need to find fg(x). We can do this by composing the functions, which means replacing x in g(x) with the expression for f(x):

fg(x) = f(g(x)) = f(2x + 1) = (2x + 1)^2 - 4

Now we can solve the inequality fg(x) > 0:

(2x + 1)^2 - 4 > 0

(2x + 1)^2 > 4

2x + 1 > 2 or 2x + 1 < -2

x > -1.5 or x < -2.5

So the values of x that make fg(x) > 0 are x > -1.5 or x < -2.5.