Question

$$\text { If } g(x)=\frac{2}{3} x-8, \text { find } g\left(\frac{3}{2} x+12\right)$$

Short answer

g\left(\frac{3}{2}x + 12\right) = x

Step-by-step solution

- Substitute the Inner Function

Identify the inner function to be substituted into the main function \(g(x)\). Here, the inner function is \(\frac{3}{2}x + 12\). Substitute this into \(g(x)\) to get \(g\left(\frac{3}{2}x + 12\right)\).

- Apply the Main Function

Use the definition of \(g(x) = \frac{2}{3}x - 8\). Replace \(x\) in the definition with the inner function: \(g\left(\frac{3}{2}x + 12\right) = \frac{2}{3}\left(\frac{3}{2}x + 12\right) - 8\).

- Simplify the Expression

Distribute \(\frac{2}{3}\) through the inner function: \(\frac{2}{3} \left(\frac{3}{2}x + 12\right) = \frac{2}{3} \cdot \frac{3}{2}x + \frac{2}{3} \cdot 12\). This simplifies to \(x + 8\). So, \(g\left(\frac{3}{2}x + 12\right) = x + 8 - 8\).

- Final Simplification

Combine like terms in the expression: \(x + 8 - 8\). The constants \(+ 8\) and \(- 8\) cancel out, leaving \(x\). Therefore, \(g\left(\frac{3}{2}x + 12\right) = x\).

Key concepts

Inner Function

In function composition, the inner function is the function that you first substitute into another function. Think of it as a 'function within a function.' In the exercise, we need to find what happens when you input \(\frac{3}{2}x+12\) into \(g(x)\).

Here, \(g(x) = \frac{2}{3} x - 8\), and our inner function is \(\frac{3}{2}x + 12\). We start by plugging this inner function into \(g(x)\).

So, this step essentially involves identifying which part needs to be substituted into the main function.

Substitution

Substitution is a key concept in function composition. When working with the inner function, you replace the variable in the outer function with the inner function's expression.

In our exercise, we take the inner function \(\frac{3}{2}x + 12\) and substitute it into \(g(x)\). Essentially, wherever we see an \(x\) in \(g(x)\), we replace it with \(\frac{3}{2}x + 12\).

• Original function: \(g(x) = \frac{2}{3} x - 8\)
  
• Substitute inner function: \(g\bigg(\frac{3}{2}x+12\bigg)=\frac{2}{3}\bigg(\frac{3}{2}x + 12\bigg) - 8\)

Distribution

Distribution involves multiplying each term inside a parenthesis by a term outside the parenthesis. In simple terms, it is about spreading out the multiplication across the terms within the parenthesis.

In our step-by-step solution, we distribute \( \frac{2}{3} \) to both \( \frac{3}{2}x \) and \(12\):

\[\frac{2}{3} \bigg(\frac{3}{2} x + 12\bigg) = \bigg(\frac{2}{3} \times \frac{3}{2}\bigg)x + \bigg(\frac{2}{3} \times 12\bigg) = x + 8 \]

After this step, the expression simplifies, allowing us to move closer to the final result.

Simplification

Simplification is about combining like terms to make an equation or expression easier to read or solve. In our exercise, after distributing, we combine \(+8\bigg)\) and \(-8\):

\[\bigg(x + 8 - 8 \bigg) = x\]

In simplification, always look for opportunities to combine like terms or eliminate terms that cancel each other out. This process makes the equation much simpler and reveals the final answer more clearly:

Thus, \(g\bigg(\frac{3}{2}x + 12\bigg) = x\).