openfreebooks-bookgenerator/content/math/functions-graphs/core.html

<div class="book-prose">
  <p>
    A <strong>function</strong> is a rule that assigns exactly one output to each allowed input.
    In high school, functions help you connect equations, tables, and graphs.
  </p>

  <h2 id="what-is-a-function" class="book-prose__heading">What is a function?</h2>
  <p>
    If \(f\) is a function, then \(f(x)\) means “the output when input is \(x\).”
    Example:
  </p>
  <p class="book-formula">
    \[
    f(x)=2x+3.
    \]
  </p>
  <p>
    Then \(f(1)=5\), \(f(4)=11\), and \(f(-2)=-1\).
  </p>
  <div class="book-callout book-callout--formula" role="note">
    <p class="book-callout__label">Key idea</p>
    <p class="book-formula">
      \[
      \text{One input} \rightarrow \text{one output}.
      \]
    </p>
  </div>

  <h2 id="domain-range" class="book-prose__heading">Domain and range</h2>
  <p>
    The <strong>domain</strong> is the set of allowed inputs. The <strong>range</strong> is the set of
    outputs produced from that domain.
  </p>
  <p><strong>Example 1.</strong> \(f(x)=\frac{1}{x-2}\)</p>
  <ul>
    <li>Domain: all real \(x\) except \(x=2\).</li>
    <li>Range: all real \(y\) except \(y=0\).</li>
  </ul>
  <p><strong>Example 2.</strong> \(g(x)=\sqrt{x+1}\)</p>
  <ul>
    <li>Domain: \(x \ge -1\) (square root needs non-negative inside).</li>
    <li>Range: \(y \ge 0\).</li>
  </ul>

  <h2 id="function-vs-relation" class="book-prose__heading">Function vs relation</h2>
  <p>
    Not every relation is a function. If one input gives two outputs, it is not a function.
  </p>
  <p>
    On a graph, use the <strong>vertical line test</strong>: if any vertical line crosses the curve
    more than once, it is not a function.
  </p>

  <h2 id="graphs-basics" class="book-prose__heading">Graph basics</h2>
  <p>
    To sketch a graph:
  </p>
  <ol>
    <li>Choose key \(x\)-values and make a value table.</li>
    <li>Plot points \((x,f(x))\) on labeled axes.</li>
    <li>Connect appropriately (line, curve, or separate branches).</li>
    <li>Identify intercepts and turning features.</li>
  </ol>

  <h2 id="key-families" class="book-prose__heading">Important function families</h2>
  <h3 id="linear" class="book-prose__heading">Linear: \(y=mx+c\)</h3>
  <p>
    Straight line with slope \(m\) and \(y\)-intercept \(c\). A larger \(|m|\) means steeper line.
  </p>

  <h3 id="quadratic" class="book-prose__heading">Quadratic: \(y=ax^2+bx+c\)</h3>
  <p>
    Parabola. If \(a>0\), opens upward; if \(a<0\), opens downward.
  </p>

  <h3 id="absolute-value" class="book-prose__heading">Absolute value: \(y=|x|\)</h3>
  <p>
    V-shaped graph with vertex at the origin before transformations.
  </p>

  <h3 id="rational" class="book-prose__heading">Rational: \(y=\frac{1}{x}\)</h3>
  <p>
    Hyperbola with asymptotes \(x=0\) and \(y=0\). Graph has two separate branches.
  </p>

  <h2 id="transformations" class="book-prose__heading">Transformations</h2>
  <p>
    Start from \(y=f(x)\):
  </p>
  <ul>
    <li>\(y=f(x)+k\): shift up by \(k\).</li>
    <li>\(y=f(x-h)\): shift right by \(h\).</li>
    <li>\(y=af(x)\): vertical stretch/compression by factor \(|a|\), and reflect if \(a&lt;0\).</li>
    <li>\(y=f(bx)\): horizontal scale by factor \(\frac{1}{|b|}\), and reflect in \(y\)-axis if \(b&lt;0\).</li>
  </ul>
  <p><strong>Example.</strong> From \(y=x^2\) to \(y=(x-3)^2+2\): move right 3, up 2.</p>

  <h2 id="intercepts-and-meaning" class="book-prose__heading">Intercepts and meaning</h2>
  <ul>
    <li><strong>\(y\)-intercept:</strong> set \(x=0\).</li>
    <li><strong>\(x\)-intercepts:</strong> solve \(f(x)=0\).</li>
  </ul>
  <p>
    In applications, intercepts often represent starting value and break-even/zero-output points.
  </p>

  <h2 id="piecewise" class="book-prose__heading">Piecewise functions</h2>
  <p>
    Some rules change in different intervals. Example:
  </p>
  <p class="book-formula">
    \[
    f(x)=
    \begin{cases}
    x+2, & x&lt;1\\
    3, & x\ge 1
    \end{cases}
    \]
  </p>
  <p>
    Graph each rule only on its stated interval, and mark endpoints correctly (open/closed points).
  </p>

  <h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
  <ul>
    <li>State domain restrictions before graphing.</li>
    <li>Use a neat value table near turning points or asymptotes.</li>
    <li>Label key coordinates: intercepts, vertex, asymptote equations.</li>
    <li>Check whether a requested equation is function form or relation form.</li>
  </ul>
</div>