Express each function as a composition of two or more functions. Verify your choice of functions.

a. \(f(x) = 2^{3x - 1}\)

b. \(g(x) = \dfrac{3}{2\sin(x) - 1}\)

Solution

a. If \(p(x) = 3x - 1\) and \(q(x) = 2^{x}\), then

Thus, \(f(x) = (q \circ p)(x)\) when \(p(x) = 3x - 1\) and \(q(x) = 2^x\).

An alternate solution has \(p(x) = 3x\) and \(q(x) = 2^{x - 1}\).

In this case,

Thus, \(f(x) = (q \circ p)(x)\) when \(p(x) = 3x\) and \(q(x) = 2^{x-1}\).

Express each function as a composition of two or more functions. Verify your choice of functions.

a. \(f(x) = 2^{3x - 1}\)

b. \(g(x) = \dfrac{3}{2\sin(x) - 1}\)

Solution

b. Approach 1

If \(p(x) = 2\sin(x) - 1\) and \(q(x) = \dfrac{3}{x}\), then

Thus, \(g(x) = (q \circ p)(x)\) when \(p(x) = 2\sin(x) - 1\) and \(q(x) = \dfrac{3}{x}\).

Express each function as a composition of two or more functions. Verify your choice of functions.

a. \(f(x) = 2^{3x - 1}\)

b. \(g(x) = \dfrac{3}{2\sin(x) - 1}\)

Solution

b. Approach 2

If \(p(x) = \sin(x)\) and \(q(x) = \dfrac{3}{2x-1}\), then

Thus, \(g(x) = (q \circ p)(x)\) when \(p(x) = \sin(x)\) and \(q(x) = \dfrac{3}{2x-1}\).

Express each function as a composition of two or more functions. Verify your choice of functions.

a. \(f(x) = 2^{3x - 1}\)

b. \(g(x) = \dfrac{3}{2\sin(x) - 1}\)

Solution

b. Approach 3

We can express \(g(x) = \dfrac{3}{2\sin(x) - 1}\) as the composition of three functions.

For example, if \(p(x) = \sin(x)\), \(q(x) = 2x - 1\), and \(r(x) = \dfrac{3}{x}\), then

Thus, \(g(x) = (r \circ q \circ p)(x)\) when \(p(x) = \sin(x)\), \(q(x) = 2x - 1\), and \(r(x) = \dfrac{3}{x}\).