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Function Basics

  • Functions can accept both numbers and letters (variables) as inputs, and the process for computation doesn't change based on the nature of the input.

Computing with Functions

  • To compute sums of functions, i.e., $f + g$, follow these steps:

    • Determine Outputs:

    • Compute $f(t)$ and $g(t)$ where $t$ is substituted into both functions.

    • If $f(t) = e^t$ and $g(t) = 3t^2 + 4$, add the outputs:

      • f(t)+g(t)=et+(3t2+4)f(t) + g(t) = e^t + (3t^2 + 4)

      • This demonstrates a straightforward addition of outputs resulting in a combination of an exponential and a quadratic function.

Multiplying Functions

  • Similar to addition, the multiplication of functions, $f imes g$, involves:

    • Calculate Outputs:

    • Substitute a new variable $s$ into both functions:

      • If $f(s) = e^s$ and $g(s) = 3s^2 + 1$, then we multiply:

      • f(s)imesg(s)=esimes(3s2+1)f(s) imes g(s) = e^s imes (3s^2 + 1)

      • Distributing the multiplication may be necessary but is dependent on preference.

Arithmetical Operations on Functions

  • The key takeaway is to perform operations (addition, multiplication) on the outputs of the functions directly.

  • This method is valid regardless of whether the inputs are numerical or symbolic letters; the process is purely about output manipulation.

Composition of Functions

  • Composition involves using the output of one function as the input for another.

  • Denotation: If you have $g(f(x))$, it means to first calculate $f(x)$ and use that result in $g$.

Concrete Example of Composition

  • Define the functions:

    • Let $f(x) = ext{square root}(x)$

    • Let $g(x) = 3$.

    • Compute the composition:

    • g(f(x))=g(extsquareroot(x))=3g(f(x)) = g( ext{square root}(x)) = 3

    • This stepwise operation underlines evaluating one function and then using that result in a second one.

Example Walk-Through for Function Composition

  • Consider the functions:

    • $f(x) = |x - 4|$ (absolute value function)

    • $g(x) = 2x^2 + 1$ (quadratic function)

Performing Composition

  • Calculate $f(g(x))$:

    1. First obtain $g(x)$:

    • g(x)=2x2+1g(x) = 2x^2 + 1

    1. Next evaluate $f$ with this result:

    • $f(g(x)) = |(2x^2 + 1) - 4| = |2x^2 - 3|$

  • Reverse the operation to calculate $g(f(x))$:

    1. For $f(x)$, we first get $f(x) = |x - 4|$

    2. Then apply that into $g$:

    • g(f(x))=g(∣x−4∣)=2(∣x−4∣)2+1g(f(x)) = g(|x - 4|) = 2(|x - 4|)^2 + 1

Multiple Compositions

  • Functions can be composed multiple times:

    • For instance, composing a third function $h(x)$ can be defined as:

    • h(g(f(x)))h(g(f(x)))

    • Stepwise calculations enhance clarity: starting with the innermost function and working outward.

Suggestion for Complex Compositions

  • When designing complex function compositions, break tasks into simple ordered steps for clarity.

  • Remember the sequence: add first, square root next, log last, etc.

  • Example setup could define:

    • $g(x) = ext{square root}(x + 1)$

    • $f(x) = ext{ln}(x)$

Final Recommendations

  • Understand that compositional operations are stepwise procedures.

  • Following the order of operations methodically will provide results while aiding comprehension.

  • No class on the following day; enjoy the weekend!