Question

Problem 8.7. Let y V4- z2 and f(z) = 2x + 3. Compute the composition %3D = g(x) y = f(g(x)). Find the largest possible domain ce le of x-values so that the composition y = f(g(x)) 't |3D is defined. X-

Answer 1

f(z) = sqrt(4 - z²)

g(x) = 2x + 3

Therefore,

y = f(g(x))

= f(2x + 3)

= sqrt(4 - (2x + 3)²)

Thus,

y = sqrt(4 - (2x + 3)²)

Domain is the set of real values of x for which the function is real and defined

For y to be real and defined, the expression inside the square root must be greater than or equal to 0

This implies

4 - (2x + 3)²$\geq$ 0

=> (2x + 3)² - 4 $\leq$ 0

=> (2x + 3)² - (2)²$\leq$ 0

=> (2x + 3 + 2)(2x + 3 - 2) $\leq$ 0

=> (2x + 5)(2x + 1) $\leq$ 0

For the above inequality to be true, either of the terms (2x + 5) or (2x + 1) must be negative and the other must be positive

Therefore, the solution is

((2x + 5) $\leq$ 0 and (2x + 1) $\geq$ 0) or ((2x + 1) $\leq$ 0 and (2x + 5) $\geq$ 0)

=> (x $\leq$ - 5 / 2 and x $\geq$ - 1 / 2) or (x $\leq$ - 1 / 2 and x $\geq$ - 5 / 2)

=> (x $\leq$ - 1 / 2 and x $\geq$ ​​​​​​​ - 5 / 2) [Since no common region exists for (x $\leq$ - 5 / 2 and x $\geq$ - 1 / 2)]

Therefore,

Domain is [- 5 / 2, -1 / 2]