Algebra 2 functions

Question

Suppose that f(x) = x^2 + 3x + 2 and h(x) = x^4 − 3x^2 + 2.

Find a function g such that f ◦ g = h.

Hint: It is not g(x) = x^2 .

Answer 1

There is more than one answer to this problem:

g(x) = -x² and g(x) = (x -3)²

Here's how to proceed. Assume that, for some value of a, b, and c, that g(x) = ax² + bx + c.

Then

h(x) = (f_°g)(x) = f( g(x) ) = f( ax² + bx + c )

= a²x⁴ + 2abx³ + [a(c+3) + b² + ac] x² + (2bc +3b)x + [c(c+3) +2]

Remember that we are given that

h(x) = x⁴ + 0 x³ + -3 x² + 0 x + 2

Since this is true for all x, then the coefficients of like powers of x must agree:

a² = 1 implies that a = 1 or -1

2ab = 0 implies that b = 0

a(c+3) + b² + ac = 3 (This is called the middle term. More about this later)

2bc + 3b = 0 (no new info since we know that b=0)

c(c+3) + 2 = 2 implies that c = 0 or -3

Let us take each possibility and compute the middle term:

a c a(c+3) + b² + ac

=== === ==================

1 0 +3

-1 0 -3

1 +3 -3

-1 -3 +3

Thus, we can now say:

g(x) = -x²

g(x) = x² - 3

Answer 2

This is what is called a composition of functions. We want to find a g(x) such that f(g(x)) = h(x). The hint brings you to almost the correct answer. Using g(x) = x^2 gives you h(x) = x^4 +3x^2 + 2. The only difference between this answer and the correct answer is the sign of 3x^2. If we use g(x) = -x^2, then we fix that issue and get the right answer.

Answer 3

do you mean f(x) times g(x) = h(x)?

If so, then g(x) = h(x) / f(x)

Otherwise it might be h(x) = f(g(x)). This notation means that the "g" function is applied to x, and then the "f" function is applied to the result.

Let g(x) = -x²

If f(x) = x² + 3x + 2, the f(g(x)) = (-x²)² + 3(-x²) + 2 = x⁴ - 3x² + 2