Function Notation

Overview

Function notation is a way to represent functions using symbols and expressions. The general form of function notation is \( f(x) \), where \( f \) is the name of the function and \( x \) is the input value.

Function notation helps us clearly express mathematical operations that we perform on a function, such as evaluating, adding, or subtracting functions. It's a more concise way of dealing with functions compared to traditional algebraic expressions.

Key Concepts

  • Notation: The function \( f(x) \) represents the output of the function when the input is \( x \). For example, if \( f(x) = 2x + 3 \), then \( f(1) = 5 \) because \( 2(1) + 3 = 5 \).
  • Evaluating Functions: To evaluate a function at a specific value, simply replace \( x \) with the given input and simplify. For example, for \( f(x) = x^2 + 1 \), \( f(3) = 9 + 1 = 10 \).
  • Function Operations: Functions can be added, subtracted, multiplied, or divided. For example, given two functions \( f(x) \) and \( g(x) \), you can write \( (f + g)(x) = f(x) + g(x) \), or \( (f \cdot g)(x) = f(x) \cdot g(x) \).

Practice Questions

  1. Evaluate \( f(x) = 3x + 4 \) at \( x = 2 \).
    Solution

    Substitute \( x = 2 \) into the function: \( f(2) = 3(2) + 4 = 6 + 4 = 10 \).

  2. If \( f(x) = x^2 + 2x \) and \( g(x) = 3x - 1 \), find \( (f + g)(x) \).
    Solution

    To find \( (f + g)(x) \), add the two functions: \( (f + g)(x) = (x^2 + 2x) + (3x - 1) = x^2 + 5x - 1 \).

  3. Find \( f(-2) \) if \( f(x) = 2x^2 - 3x + 1 \).
    Solution

    Substitute \( x = -2 \) into the function: \( f(-2) = 2(-2)^2 - 3(-2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15 \).

  4. Given \( f(x) = x^2 \) and \( g(x) = 4x + 3 \), find \( (f \cdot g)(x) \).
    Solution

    To find \( (f \cdot g)(x) \), multiply the two functions: \( (f \cdot g)(x) = (x^2)(4x + 3) = 4x^3 + 3x^2 \).

  5. Evaluate \( h(x) = x^3 - 4x \) at \( x = -1 \).
    Solution

    Substitute \( x = -1 \) into the function: \( h(-1) = (-1)^3 - 4(-1) = -1 + 4 = 3 \).

  6. If \( f(x) = 2x + 3 \) and \( g(x) = x^2 - 1 \), find \( (f - g)(x) \).
    Solution

    To find \( (f - g)(x) \), subtract \( g(x) \) from \( f(x) \): \( (f - g)(x) = (2x + 3) - (x^2 - 1) = -x^2 + 2x + 4 \).

  7. What is the value of \( f(0) \) if \( f(x) = 5x - 3 \)?
    Solution

    Substitute \( x = 0 \) into the function: \( f(0) = 5(0) - 3 = -3 \).

  8. Given \( f(x) = 2x - 5 \) and \( g(x) = x^2 + 4 \), find \( (f \cdot g)(x) \).
    Solution

    To find \( (f \cdot g)(x) \), multiply the two functions: \( (f \cdot g)(x) = (2x - 5)(x^2 + 4) = 2x^3 + 8x - 5x^2 - 20 \).

  9. If \( f(x) = x^2 - 4 \) and \( g(x) = 2x + 3 \), find \( (f - g)(x) \).
    Solution

    To find \( (f - g)(x) \), subtract \( g(x) \) from \( f(x) \): \( (f - g)(x) = (x^2 - 4) - (2x + 3) = x^2 - 2x - 7 \).

  10. What is the function value \( f(3) \) for \( f(x) = 4x^2 - x + 2 \)?
    Solution

    Substitute \( x = 3 \) into the function: \( f(3) = 4(3)^2 - 3 + 2 = 36 - 3 + 2 = 35 \).

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