Finding the Range of a Function (Algebraic Methods)

What Is “Range”?

• The set of all possible output (dependent-variable) values $y$ a function $y = f(x)$ can take.
• Formal description: $\text{Range}(f)={\,y\mid \exists\,x\in\text{Domain}(f)\;\text{s.t.}\;y=f(x)\,}$
• Historically in statistics: “highest − lowest” value in a data set. In functions, the idea generalizes to every achievable output, not a single difference.

Why Some Values Are Excluded

• An output is impossible if no admissible $x$ produces it.
  • Function might be bounded (e.g.
  $f(x)=-x^4+1$ can never exceed $1$).
  • Vertical asymptotes / discontinuities can leave gaps (e.g. a rational function skips the horizontal asymptote value).
  • Radicals or logarithms after inversion may restrict $y$ (square-root needs a non-negative radicand, $\ln(\cdot)$ needs positive input, etc.).

Core Algebraic Strategy

1. Write the function in the form $y = f(x)$.
2. Isolate $x$ in terms of $y$, giving $x = g(y)$ (the “inverse relation”).
3. Determine the domain of $g(y)$ (all $y$ for which $g(y)$ is defined).
   • Any restriction on $y$ here becomes a restriction on the range of $f$.
4. Translate the domain of $g$ into the range of $f$.
5. Check special or excluded $x$ values in the original function to ensure their corresponding $y$ outputs are removed if needed.
6. Graph if algebraic inversion is impractical; blend graphical intuition with analytic bounds.

Worked Example 1 – Quadratic Shift

Given $f(x)=x^2+1$

• Step 1 $y = x^2 + 1$
• Step 2 Solve for $x$: $x^2 = y-1\quad\Rightarrow\quad x=\pm\sqrt{y-1}$
• Step 3 Domain requirement of $g(y)=\pm\sqrt{y-1}$: $y-1\ge0\;\Rightarrow\;y\ge1$
• Range $[1,\infty)$
  • Test value: $y=5$ gives $x=\pm2$, confirming inclusion.

Worked Example 2 – Absolute Value

Function $f(x)=|x|$

• Intuition: output never negative.
• Formal proof:
  • Pick arbitrary $y\ge0$.
  • Choose $x=y$ (or $x=-y$) so $|x|=y$.
• Range $[0,\infty)$.
• Practice check: Solve $|x|=12$ ⇒ $x=12,\,-12$ (two symmetric pre-images).

Worked Example 3 – Quartic with Reflection & Shift

$f(x)=-x^4+1$

• Domain: all reals (polynomial).
• Observations:
  • $x^4\ge0\Rightarrow -x^4\le0$.
  • Maximum occurs at $x=0$: $f(0)=1$.
• Algebraic confirmation:

1. $y=-x^4+1$
2. $y-1=-x^4$ ⇒ $1-y=x^4$
3. Need $1-y\ge0\Rightarrow y\le1$.
4. Fourth-root: $x=\pm\sqrt[4]{1-y}$ defines real $x$ for every $y\le1$.

• Range $( -\infty,\,1 ]$ (open at $-\infty$, closed at $1$).

Quick-Reference Table – Common Families

• Absolute value: $f(x)=|x|\quad\Rightarrow\quad [0,\infty)$
• Logarithm (any base >1 or <1): $f(x)=\log_a x\quad\Rightarrow\quad (-\infty,\infty)$
• Exponential: f(x)=a^x,\; a>0,\,a\neq1\quad\Rightarrow\quad (0,\infty)

General Tips & Pitfalls

• Always confirm the domain first; excluded $x$ can knock holes out of the range.
• When inversion gives an even root, impose $\text{radicand}\ge0$.
• For rational functions, set denominator $\neq0$ before solving for $x$.
• Infinity endpoints are always written with parentheses: $(k,\infty)\text{ or }(-\infty,k)$.
• If the inverse relation involves $\ln y$, remember y>0.
• If stuck: sketch a quick graph (monotonicity, asymptotes, extrema).

Conceptual Connections & Real-World Relevance

• Range analysis underpins solving equations: knowing reachable outputs tells you whether a target value is feasible.
• In optimization, extrema (max/min) are edges of the range; many engineering and economic problems reduce to identifying these bounds.
• Control systems & physics: ensuring output stays within safe range prevents saturation/overload.
• Data science: understanding domain-range transformations (e.g.
  activation functions in neural networks) is crucial for gradient behaviors.

Summary Algorithm (4-Step Version Recap)

1. Declare $y=f(x)$.
2. Express $x$ explicitly: $x=g(y)$.
3. Find all $y$ where $g(y)$ is defined and original domain conditions hold.
4. State range, adjusting for unattainable outputs.

Key Terms

• Range: set of possible $y$ values.
• Domain: allowed $x$ inputs.
• Inverse relation: expression of $x$ in terms of $y$ (may not be a formal function if it fails vertical-line test).
• Bounded: function has upper and/or lower finite limits.
• Asymptote: line a function approaches but never meets; its $y$-value may be excluded from the range.