2.1 Function language, evaluation, domain, and range

A function formalizes a dependable input-output relationship. Each allowable input must produce exactly one output, although different inputs may produce the same output.

Function. A rule that assigns exactly one output to every input in its domain.

If a function maps a set \(A\) to a set \(B\), write \(f:A\to B\). The set \(A\) is the domain and \(B\) is the codomain. The value assigned to \(x\) is \(f(x)\), read “\(f\) of \(x\).” The set of outputs actually produced is the range. The range is contained in the codomain but does not always equal it.

2.1.1 Evaluating functions and recognizing graphs

To evaluate a function, replace every occurrence of its variable with the specified input. Parentheses are essential when the input is negative or algebraic. For \(f(x)=x^2-3x\), for example,

\[f(-2)=(-2)^2-3(-2)=10.\]

An input \(a\) produces the point \((a,f(a))\) on the graph. A graph represents a function of \(x\) exactly when every vertical line intersects it at most once. This vertical line test checks whether one input has been assigned more than one output. A circle fails the test, while a vertical parabola such as \(y=x^2\) passes.

Worked Example: Evaluating a flow function

The flow through a control setting is modelled by \(Q(s)=2.5s^2+4\), where \(Q\) is in litres per second. At \(s=3\),

\[Q(3)=2.5(3)^2+4=26.5\ \text{L/s}.\]

The ordered pair \((3,26.5)\) lies on the graph.

Try It. Evaluate \(Q(-2)\) for the same algebraic rule.
Check Your Work \(Q(-2)=2.5(-2)^2+4=14\) L/s. Whether a negative setting is physically permitted is a separate domain question.

Worked Example: Evaluating a difference quotient input

For \(f(x)=1/(x+1)^2\),

\[f(x+h)=\frac{1}{(x+h+1)^2}.\]

The entire expression \(x+h\) replaces the original input variable.

Try It. If \(g(x)=3x^2-1\), find \(g(a+h)\).
Check Your Work \(g(a+h)=3(a+h)^2-1\).

2.1.2 Domain, codomain, and range

Domain. The set of inputs for which a function is defined.

Range. The set of outputs a function actually produces.

The algebraic domain is found by excluding operations that are undefined over the real numbers. A denominator cannot equal zero. The radicand of an even root must be nonnegative, and it must be positive when that root appears in a denominator. Logarithmic inputs, introduced later, must be positive. Physical models may add restrictions such as \(t\ge0\), \(Q\ge0\), or a maximum equipment setting.

Range is often easiest to determine from a graph. The function \(g(x)=|x|\) has domain \(\mathbb R\) and range \([0,\infty)\). A quadratic opening upward has a minimum output at its vertex. A reciprocal function may exclude a horizontal value. State range restrictions on outputs, not inputs.

Worked Example: Finding a radical domain

For

\[h(x)=\frac{2+x}{\sqrt{x(x+3)}},\]

the denominator requires \(x(x+3)>0\). A sign chart gives \(x<-3\) or \(x>0\), so the domain is \((-\infty,-3)\cup(0,\infty)\).

Try It. Find the domain of \(k(x)=1/\sqrt{x-5}\).
Check Your Work The denominator requires \(x-5>0\), so the domain is \((5,\infty)\).

Worked Example: Interpreting the range of a storage model

A tank-depth model is \(D(t)=4-0.2t\) for \(0\le t\le20\) min. The depth decreases from 4 m to 0 m. Its physical domain is \([0,20]\) and its range is \([0,4]\) m.

Try It. Find the range of \(V(t)=10+3t\) for \(0\le t\le8\).
Check Your Work \(V\) increases from 10 to 34, so the range is \([10,34]\).

Applied Problem: Restricting a reservoir model. Reservoir volume is modelled by \(V(t)=120-4.5t\) megalitres, where \(t\) is measured in days. Determine the largest physical domain for which time and volume are nonnegative, and state the corresponding range.

Check Your Work Require \(t\ge0\) and \(120-4.5t\ge0\). Thus \(0\le t\le120/4.5=26.67\) days. The physical domain is approximately \([0,26.67]\) days and the range is \([0,120]\) ML.

2.1.3 Practice Problems

  1. For \(f(x)=x^2-4x+1\), find \(f(3)\).
    Check Your Work \(f(3)=9-12+1=-2\).
  2. If \(g(x)=2/(x-1)\), find \(g(a+h)\).
    Check Your Work \(g(a+h)=2/(a+h-1)\).
  3. Does \(x=y^2\) define \(y\) as a function of \(x\) over all real values?
    Check Your Work No. For most positive \(x\), the relation has two outputs, \(y=\pm\sqrt{x}\).
  4. Find the domain of \(\sqrt{12-3x}\).
    Check Your Work \(12-3x\ge0\), so \(x\le4\).
  5. State the domain and range of \(f(x)=(x-2)^2+1\).
    Check Your Work Domain \(\mathbb R\); range \([1,\infty)\).
  6. A flow model is \(Q(v)=0.8v+6\). Find \(Q(25)\) and interpret it if \(Q\) is measured in L/s.
    Check Your Work \(Q(25)=26\) L/s.
  7. A water-depth function is \(d(t)=3.6-0.12t\) for \(0\le t\le30\). State its range.
    Check Your Work The depth falls from 3.6 m to 0 m, so the range is \([0,3.6]\) m.
  8. Find the physical domain of \(A(d)=\sqrt{25-d^2}\) when \(d\) is a nonnegative pipe dimension.
    Check Your Work \(25-d^2\ge0\) and \(d\ge0\), giving \(0\le d\le5\).