Which Of The Following Graphs Have The Same Domain ?

When studying functions and graphs in algebra or pre-calculus, one common question is: “Which of the following graphs have the same domain?” Understanding this concept is essential for graph analysis, problem-solving, and preparing for standardized tests. This article will guide you through everything you need to know to answer this question confidently.

Understanding the Concept of Domain

Before comparing graphs, let’s clarify what domain means:

• The domain of a function is the set of all input values (x-values) for which the function is defined.
• In simpler terms, it tells you all the x-values you can plug into a function without breaking mathematical rules.

Examples of Common Domain Restrictions:

1. Square roots – You cannot take the square root of a negative number in real numbers: $$f(x) = \sqrt{x} \implies x \geq 0$$f(x)=x​⟹x≥0
2. Division by zero – You cannot divide by zero: $$f(x) = \frac{1}{x-2} \implies x \neq 2$$f(x)=x−21​⟹x=2
3. Logarithms – The argument of a logarithm must be positive: $$f(x) = \ln(x) \implies x > 0$$f(x)=ln(x)⟹x>0

All other x-values are usually valid for functions unless explicitly restricted.

Step 1: Reading Graphs to Determine the Domain

When a problem asks “Which of the following graphs have the same domain?”, your first step is to carefully examine each graph along the x-axis.

Tips to find a graph’s domain:

1. Look horizontally – The domain corresponds to the horizontal span of the graph.
2. Identify breaks or holes – Open circles, vertical asymptotes, or missing segments indicate restrictions.
3. Check endpoints – Closed circles indicate the graph includes that x-value; open circles indicate it does not.

Example:

• A parabola like $y = x^2$y=x2 extends infinitely left and right. Domain: all real numbers.
• A square root function like $y = \sqrt{x-3}$y=x−3​ starts at $x = 3$x=3 and extends right. Domain: $x \geq 3$x≥3.

Step 2: Comparing Multiple Graphs

Once you know each graph’s domain:

1. Write the domain in interval notation – This helps you compare easily.
   • All real numbers: $(-\infty, \infty)$(−∞,∞)
   • Starting at a number: $[3, \infty)$[3,∞)
   • Excluding points: $(-\infty, 2) \cup (2, \infty)$(−∞,2)∪(2,∞)
2. Match identical intervals – Graphs with the same interval have the same domain.

Example Question:

Suppose we have four graphs:

• Graph A: $y = x^2$y=x2 → Domain: $(-\infty, \infty)$(−∞,∞)
• Graph B: $y = \sqrt{x-1}$y=x−1​ → Domain: $[1, \infty)$[1,∞)
• Graph C: $y = x^3$y=x3 → Domain: $(-\infty, \infty)$(−∞,∞)
• Graph D: $y = \frac{1}{x+2}$y=x+21​ → Domain: $(-\infty, -2) \cup (-2, \infty)$(−∞,−2)∪(−2,∞)

✅ Same domain: Graph A and Graph C (all real numbers)

Step 3: Recognizing Patterns in Graphs

Certain types of functions usually have the same domains:

1. Polynomial functions (linear, quadratic, cubic, quartic, etc.)
   • Domain: all real numbers, unless restricted.
   • Examples: $y = x$y=x, $y = x^2$y=x2, $y = x^3 – 2x$y=x3−2x
2. Exponential functions
   • Domain: all real numbers
   • Example: $y = 2^x$y=2x
3. Rational functions
   • Domain: all x-values except where the denominator is zero
   • Example: $y = \frac{1}{x-5}$y=x−51​ → Domain: $x \neq 5$x=5
4. Square root and logarithmic functions
   • Domain is restricted to ensure the inside of the root or log is valid
   • Example: $y = \sqrt{x+3}$y=x+3​ → Domain: $x \geq -3$x≥−3

Step 4: Common Pitfalls

1. Assuming all graphs extend infinitely – Not true for square roots, piecewise functions, and rationals with restrictions.
2. Ignoring open circles or asymptotes – These indicate x-values are excluded from the domain.
3. Confusing domain with range – Domain is about x-values, range is about y-values.

Step 5: Practical Strategy to Answer Multiple-Choice Questions

1. Check horizontal coverage – For each graph, note where it starts and ends along the x-axis.
2. Note exceptions – Identify holes, asymptotes, or gaps.
3. Convert to intervals – Makes comparison easy.
4. Look for matching intervals – These graphs have the same domain.

Example:

Graph	Domain
A	$(-\infty, \infty)$(−∞,∞)
B	$[0, \infty)$[0,∞)
C	$(-\infty, \infty)$(−∞,∞)
D	$(-\infty, 2) \cup (2, \infty)$(−∞,2)∪(2,∞)

✅ Answer: Graph A and C

Why Understanding Domains Matters

• Graphing accurately – Helps in plotting correct points and lines.
• Solving equations – Avoids invalid x-values that make functions undefined.
• Real-world modeling – Domains often represent practical constraints like time, distance, or quantity.

Summary

To determine “Which of the following graphs have the same domain?”, follow these steps:

1. Examine each graph horizontally.
2. Identify any breaks, holes, or endpoints.
3. Write the domain in interval notation.
4. Compare all intervals to find matches.

Key Takeaways:

• Polynomial and exponential graphs usually have the same domain (all real numbers).
• Square roots, logarithms, and rational functions often have restricted domains.
• Always pay attention to asymptotes, holes, and open circles.