Question

To find the value of "a" in the function g(x) = x^3 + a when it's composed with f(x) = √(x + 2) to form h(x) = √(x + 2)^3 - 2, you should substitute the function g(x) into the function f(x) and simplify. Here's how it's done:

h(x) = f(g(x))
h(x) = √(x^3 + a + 2) - 2

Now, since h(x) = √(x + 2)^3 - 2, we can equate the two expressions:

√(x^3 + a + 2) - 2 = √(x + 2)^3 - 2

Now, we can eliminate the common term "-2" on both sides:

√(x^3 + a + 2) = √(x + 2)^3

Now, to solve for "a," we'll square both sides:

x^3 + a + 2 = (x + 2)^3

Expand the right side:

x^3 + a + 2 = x^3 + 6x^2 + 12x + 8

Now, subtract x^3 from both sides to isolate "a":

a + 2 = 6x^2 + 12x + 8

Subtract 2 from both sides to find the value of "a":

a = 6x^2 + 12x + 6

So, the value of "a" is a quadratic expression: a = 6x^2 + 12x + 6.

Answer 1

To find the value of "a" in the given composition of functions, substitute g(x) into f(x), equate the expressions for h(x), simplify, and solve for "a".

Step-by-step explanation:

To find the value of "a" in the function g(x) = x^3 + a when it's composed with f(x) = √(x + 2) to form h(x) = √(x + 2)^3 - 2, we can start by substituting g(x) into f(x) and simplifying.

Then, we equate the expressions for h(x) obtained from both compositions and simplify further.

Finally, by comparing the coefficients of the equation, we can solve for "a" and find that a = 6x^2 + 12x + 6.