Finding the Value of k in Function Composition

This problem asks us to find the value of a constant 'k' when the composition of two linear functions, f(x) and g(x), is commutative, meaning fog(x) = gof(x).

Understanding Function Composition

Function composition means applying one function to the result of another function. For two functions f(x) and g(x):

• fog(x) means f(g(x)) - we substitute the function g(x) into f(x).
• gof(x) means g(f(x)) - we substitute the function f(x) into g(x).

Given Functions

We are given the following linear functions:

• f(x) = 4x + 1
• g(x) = kx + 2

And the condition: fog(x) = gof(x).

Calculating fog(x)

To find fog(x), we substitute g(x) into f(x):

\(\text{fog(x)} = \text{f(g(x))}\)

\(\text{fog(x)} = \text{f(kx + 2)}\)

Now, replace 'x' in f(x) with '(kx + 2)':

\(\text{fog(x)} = 4(\text{kx + 2}) + 1\)

\(\text{fog(x)} = 4\text{kx} + 8 + 1\)

\(\text{fog(x)} = 4\text{kx} + 9\)

Calculating gof(x)

To find gof(x), we substitute f(x) into g(x):

\(\text{gof(x)} = \text{g(f(x))}\)

\(\text{gof(x)} = \text{g(4x + 1)}\)

Now, replace 'x' in g(x) with '(4x + 1)':

\(\text{gof(x)} = \text{k(4x + 1)} + 2\)

\(\text{gof(x)} = 4\text{kx} + \text{k} + 2\)

Solving for k using the Condition fog(x) = gof(x)

We are given that fog(x) = gof(x). Let's set the expressions we found equal to each other:

\(4\text{kx} + 9 = 4\text{kx} + \text{k} + 2\)

Now, we need to solve this equation for the value of k. Notice that the term \(4\text{kx}\) appears on both sides of the equation. We can subtract \(4\text{kx}\) from both sides:

\(4\text{kx} + 9 - 4\text{kx} = 4\text{kx} + \text{k} + 2 - 4\text{kx}\)

This simplifies to:

\(9 = \text{k} + 2\)

To isolate k, subtract 2 from both sides of the equation:

\(9 - 2 = \text{k} + 2 - 2\)

This gives us:

\(7 = \text{k}\)

So, the value of k that satisfies the condition fog(x) = gof(x) is 7.

Summary of Steps to Find k

1. Write down the given functions, f(x) and g(x).
2. Calculate the composite function fog(x) = f(g(x)).
3. Calculate the composite function gof(x) = g(f(x)).
4. Set fog(x) equal to gof(x) based on the problem condition.
5. Solve the resulting equation for the unknown constant k.

Revision Table: Function Composition

Additional Information: Linear Function Composition

When composing two linear functions, say \(f(x) = ax + b\) and \(g(x) = cx + d\):

• fog(x) = f(cx + d) = a(cx + d) + b = acx + ad + b
• gof(x) = g(ax + b) = c(ax + b) + d = acx + cb + d

For fog(x) = gof(x), we need:

\(acx + ad + b = acx + cb + d\)

Subtracting \(acx\) from both sides gives:

\(ad + b = cb + d\)

This is the general condition for the composition of two linear functions \(ax+b\) and \(cx+d\) to be commutative. In our specific problem, \(a=4\), \(b=1\), \(c=k\), \(d=2\). Plugging these values into the general condition:

\(4(2) + 1 = k(1) + 2\)

\(8 + 1 = k + 2\)

\(9 = k + 2\)

\(k = 9 - 2\)

\(k = 7\)

This confirms the result obtained by directly calculating fog(x) and gof(x) for the given functions.