Question

Given that f(x)=x2−4 and g(x)=x+3 , what are all the values of x for which f(g(x))=0 ?

Answer 1

The values of x for which f(g(x)) = 0 are x = -1 and x = -5, which are obtained by composing the two given functions, setting the resulting quadratic equation to zero, and solving for x using the quadratic formula.

Step-by-step explanation:

To find all the values of x for which f(g(x)) = 0 given that f(x) = x2 - 4 and g(x) = x + 3, we need to compose the functions and solve the resulting equation.

First, we find f(g(x)) by plugging g(x) into f(x):

• f(g(x)) = f(x + 3) = (x + 3)2 - 4

Next, we expand and simplify:

• f(g(x)) = x2 + 6x + 9 - 4
• f(g(x)) = x2 + 6x + 5

Now, we set this expression equal to zero:

• x2 + 6x + 5 = 0

We can solve this quadratic equation using the quadratic formula x = [-b ± √(b2 - 4ac)]/(2a), where a = 1, b = 6, and c = 5:

x = [-6 ± √(62 - 4(1)(5))]/(2(1))

x = [-6 ± √(36 - 20)]/2

x = [-6 ± √16]/2

x = [-6 ± 4]/2

So, the values of x are:

• x = (-6 + 4)/2 = -1
• x = (-6 - 4)/2 = -5

Thus, the values of x for which f(g(x)) = 0 are x = -1 and x = -5.