Find The Value Of F 1

Finding the Value of f(1): A Comprehensive Guide

Determining the value of f(1) depends entirely on the definition of the function f. There's no single answer; the solution hinges on the specific function provided. This article will explore various scenarios, providing a step-by-step guide to finding f(1) for different types of functions, including algebraic, piecewise, recursive, and those involving limits and derivatives. We will also cover strategies for handling undefined cases and common pitfalls to avoid.

Understanding Function Notation

Before diving into specific examples, let's solidify our understanding of function notation. The notation f(x) represents a function named 'f' that takes an input value 'x' and produces an output value. f(1) specifically asks for the output of function 'f' when the input is 1. Think of it as a machine: you input 1, the function processes it according to its definition, and the output is f(1).

Algebraic Functions

This is the most common type of function you'll encounter. An algebraic function is defined using algebraic operations like addition, subtraction, multiplication, division, and exponentiation.

Example 1: Find f(1) if f(x) = 2x + 3.

This is a simple linear function. To find f(1), we substitute x = 1 into the function:

f(1) = 2(1) + 3 = 2 + 3 = 5

Therefore, f(1) = 5.

Example 2: Find f(1) if f(x) = x² - 4x + 7.

This is a quadratic function. Substituting x = 1:

f(1) = (1)² - 4(1) + 7 = 1 - 4 + 7 = 4

Therefore, f(1) = 4.

Example 3: Find f(1) if f(x) = (x³ + 2x)/(x - 1).

This is a rational function. Substituting x = 1 directly leads to division by zero, which is undefined.

f(1) = (1³ + 2(1))/(1 - 1) = 3/0 This is undefined. The function f(x) has a vertical asymptote at x = 1.

Piecewise Functions

Piecewise functions are defined differently over different intervals. To find f(1), we must determine which piece of the function applies to the input x = 1.

Example 4: Find f(1) if:

f(x) = { 
x² + 1,  if x < 1
2x - 1, if x ≥ 1 
}

Since 1 ≥ 1, we use the second piece of the function:

f(1) = 2(1) - 1 = 1

Therefore, f(1) = 1.

Recursive Functions

Recursive functions define their output in terms of previous outputs. They often require an initial condition or base case.

Example 5: Find f(1) if f(n) = f(n-1) + 2, with f(0) = 1.

This is a simple recursive function. We start with the base case f(0) = 1. Then:

f(1) = f(1-1) + 2 = f(0) + 2 = 1 + 2 = 3

Therefore, f(1) = 3.

More complex recursive functions might require iterative calculations or use of mathematical induction to find f(1).

Functions Involving Limits

Sometimes, f(1) might not be directly defined, but we can find its value using limits. This is often the case when dealing with functions that have removable discontinuities.

Example 6: Find the value of lim (x→1) f(x) if f(x) = (x² - 1)/(x - 1).

We cannot substitute x = 1 directly because it leads to 0/0, an indeterminate form. However, we can factor the numerator:

f(x) = (x - 1)(x + 1)/(x - 1) = x + 1 (for x ≠ 1)

Now, we can take the limit:

lim (x→1) (x + 1) = 1 + 1 = 2

Therefore, even though f(1) is undefined, the limit as x approaches 1 is 2. This suggests a removable discontinuity at x = 1.

Functions Involving Derivatives

Derivatives represent the instantaneous rate of change of a function. Finding f(1) might involve using the derivative if the problem relates to tangent lines or rates of change.

Example 7: If f'(x) = 2x and f(0) = 3, find f(1).

We need to integrate f'(x) to find f(x):

∫2x dx = x² + C (where C is the constant of integration)

We use the initial condition f(0) = 3 to find C:

3 = (0)² + C => C = 3

So, f(x) = x² + 3. Now we can find f(1):

f(1) = (1)² + 3 = 4

Therefore, f(1) = 4.

Handling Undefined Cases and Common Pitfalls

Several scenarios can lead to undefined values for f(1):

• Division by zero: As seen in Example 3, this is a common cause. Check carefully for any values of x that would result in division by zero.
• Square roots of negative numbers: Ensure the input to any square root is non-negative.
• Logarithms of non-positive numbers: The logarithm function is only defined for positive inputs.
• Trigonometric functions with undefined values: Certain trigonometric functions are undefined at specific angles.

Common Pitfalls:

• Incorrect substitution: Carefully substitute the value of x = 1 into the function. Double-check your calculations.
• Ignoring piecewise definitions: Always check which piece of the function applies to the input x = 1 for piecewise functions.
• Forgetting constants of integration: When integrating to find f(x) from f'(x), remember to include the constant of integration and use any given initial conditions to solve for it.
• Misinterpreting limits: Understand the difference between the function's value at a point and the limit of the function as x approaches that point.

Conclusion

Finding the value of f(1) requires careful attention to the specific function's definition. This article provided a range of examples and strategies for handling different function types, including algebraic, piecewise, recursive, and those involving limits and derivatives. By understanding function notation, paying close attention to detail, and being aware of potential undefined cases, you can confidently tackle a wide variety of problems involving the evaluation of f(1). Remember to always double-check your work and systematically approach the problem to avoid common errors. Mastering this skill is fundamental for success in calculus and various other mathematical fields.