Problem 18 - Precalculus

Question

Use the given function \(f\) to find and simplify the following: \- \(f(3)\) \- \(f(4 x)\) \- \(f(x-4)\) \- \(f(-1)\) \- \(4 f(x)\) -\(f(x)-4\) \- \(f\left(\frac{3}{2}\right)\) \- \(f(-x)\) \- \(f\left(x^{2}\right)\) $$f(x)=0$$

Step-by-Step Solution

Answer

All function evaluations and transformations result in 0, except for \( f(x)-4 \), which is -4.

Step 1: Evaluate f(3)

Given the function \( f(x) = 0 \), we need to evaluate \( f(3) \). Since the function is constant and always equals zero, substitute \( x = 3 \) into the function: \( f(3) = 0 \).

Step 2: Evaluate f(4x)

Substitute \( 4x \) for \( x \) in \( f(x) = 0 \). Since the function \( f(x) = 0 \) is constant, it yields \( f(4x) = 0 \).

Step 3: Evaluate f(x-4)

Substitute \( x-4 \) for \( x \) in the function \( f(x) = 0 \). The constant function will still evaluate to \( f(x-4) = 0 \).

Step 4: Evaluate f(-1)

Substitute \( -1 \) for \( x \) in the function \( f(x) = 0 \). The constant function evaluates to \( f(-1) = 0 \).

Step 5: Compute 4f(x)

Since \( f(x) = 0 \), multiplying it by 4 gives \( 4f(x) = 4 imes 0 = 0 \).

Step 6: Compute f(x) - 4

With \( f(x) = 0 \), calculate \( f(x) - 4 = 0 - 4 = -4 \).

Step 7: Evaluate f(3/2)

Substitute \( \frac{3}{2} \) for \( x \) in \( f(x) = 0 \). The function evaluates to \( f(3/2) = 0 \).

Step 8: Evaluate f(-x)

Substitute \( -x \) for \( x \) in \( f(x) = 0 \). The constant function results in \( f(-x) = 0 \).

Step 9: Evaluate f(x²)

Substitute \( x^2 \) for \( x \) in \( f(x) = 0 \). The function evaluates as \( f(x^2) = 0 \).

Step 10: Evaluate f(x) = 0

The provided function is already \( f(x) = 0 \), so no additional evaluation is necessary.

Key Concepts

Function Evaluation

Function evaluation refers to the process of determining the output of a function when given a specific input. In mathematics, especially precalculus, evaluating functions allows us to understand how different inputs affect the output.

When a function is provided, such as the constant function in our example, which is given by \( f(x) = 0 \), evaluating it becomes straightforward. Here, no matter what value of \( x \) you substitute, the result is always zero.

For example:

• \( f(3) = 0 \)
• \( f(4x) = 0 \)
• \( f(x-4) = 0 \)
• \( f\left(\frac{3}{2}\right) = 0 \)

This simplicity stems from the nature of constant functions, where the output does not depend on the input.

Substitution in Functions

Substitution involves replacing a variable with another expression or value. It is a common technique used to evaluate or manipulate functions.

In our function \( f(x) = 0 \), regardless of the substitution made, the function remains at zero.

Here are some examples:

• Substituting \(-1\) gives \( f(-1) = 0 \).
• Substituting \(4x\) results in \( f(4x) = 0 \).
• Replacing \(x\) with \(x-4\), we get \( f(x-4) = 0 \).

In each case, we replace \( x \) with another term, yet because our function is constant, the output remains unaffected. Understanding substitution is key to mastering function evaluations in algebra and precalculus.

Algebraic Manipulation

Algebraic manipulation refers to using algebraic rules and operations to simplify or alter expressions and functions for easier calculation and comprehension.

With constant functions like \( f(x) = 0 \), algebraic manipulation can involve processes such as multiplication or subtraction:

• Multiplying by a scalar: \( 4f(x) = 4 \times 0 = 0 \).
• Subtracting a value: \( f(x) - 4 = 0 - 4 = -4 \).

These operations show how versatile and simple algebraic manipulation can be when applied to constant functions. You perform regular arithmetic with the constant value, maintaining consistency in outcomes.

Precalculus

Precalculus lays a significant groundwork for calculus by introducing concepts and techniques for analyzing functions and equations. Constant functions, like \( f(x) = 0 \), present a basic yet crucial concept within precalculus.

Understanding these functions helps build a clear idea of function behavior and paves the way to more advanced topics such as limits and derivatives in calculus.

In precalculus, you'll often deal with:

• The behavior and properties of functions.
• Methods like function evaluation and substitution.
• Simplification through algebraic manipulation.

Grasping these foundational skills is essential for smoothly transitioning into calculus, where the complexity of functions increases and requires more in-depth analysis of changes and trends in their behavior.