Question

If \(f(x)=4 x+3,\) find \(f\left(\frac{x-3}{4}\right)\)

Short answer

The answer is \( x \).

Step-by-step solution

Understand the function

First, understand the given function. The function provided is: \[ f(x) = 4x + 3 \]

Substitute into the function

We need to substitute \( \frac{x-3}{4} \) into the function \( f(x) \). This means replacing every occurrence of \( x \) in \( f(x) \) with \( \frac{x-3}{4} \).

Perform the substitution

Substitute \( \frac{x-3}{4} \) into \( f(x) \): \[ f\left( \frac{x-3}{4} \right) = 4 \left( \frac{x-3}{4} \right) + 3 \]

Simplify the expression

Now simplify the right-hand side: \[ f\left( \frac{x-3}{4} \right) = 4 \cdot \frac{x-3}{4} + 3 \] Upon simplifying: \[ f\left( \frac{x-3}{4} \right) = x-3+3 \]Finally, combine like terms to get:\[ f\left( \frac{x-3}{4} \right) = x \]

Key concepts

Evaluating Functions

Evaluating functions is a fundamental concept in algebra. It means determining the output of a function for a given input. Here, we have the function:
\( f(x) = 4x + 3 \)
To evaluate the function for a specific value, substitute that value for the variable \( x \). For example, if we want to find \( f(2) \), we replace \( x \) with 2:
\( f(2) = 4 \times 2 + 3 = 8 + 3 = 11 \) Understanding this basic concept allows us to apply it to more complex situations, like function substitution.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions using algebraic rules. In our problem, we started with the function \( f(x) = 4x + 3 \) and needed to find \( f \left( \frac{x-3}{4} \right) \). This involved a few steps:

• Substitute \( \frac{x-3}{4} \) into the function.


• Simplify the resulting expression.

We substitute: \[ f \left( \frac{x-3}{4} \right) = 4 \left( \frac{x-3}{4} \right) + 3 \]
Simplifying inside the parentheses: \[ = 4 \times \frac{x-3}{4} + 3 \] Cancelling out the 4s: \[ = x - 3 + 3 \] Combining like terms gives us: \( f \left( \frac{x-3}{4} \right) = x \). By mastering algebraic manipulation, you can tackle more complex algebra problems easily.

Function Composition

Function composition involves applying one function to the results of another. It’s like putting one process inside another. If \( f \) and \( g \) are two functions, then the composite function \( f(g(x)) \) means:

• First apply \( g(x) \)
• Then use the output of \( g(x) \) as the input for \( f \)

In our exercise, we found \( f \left( \frac{x-3}{4} \right) \). Here, \( \frac{x-3}{4} \) is effectively \( g(x) \), and we are applying the function \( f \) to \( g(x) \). To solve this: 1. Identify the inner function \( g(x) = \frac{x-3}{4} \).
2. Apply \( f \) to the result of \( g(x) \): \( f \left( \frac{x-3}{4} \right) = 4 \left( \frac{x-3}{4} \right) + 3 \) 3. Simplify to find the final expression. Understanding function composition is critical in higher algebra, as it helps in breaking down complex functions into simpler, manageable parts.