Domain and Range

Finding domain and range of functions

Domain and Range

Definitions

Domain: The set of all possible input values (x-values)

Range: The set of all possible output values (y-values)

Finding Domain

Ask: "What x-values can I put into this function?"

Common restrictions:

  • Cannot divide by zero
  • Cannot take square root of negative numbers (in real numbers)

Example 1: f(x)=2x+3f(x) = 2x + 3

  • Domain: All real numbers

Example 2: f(x)=1x−5f(x) = \frac{1}{x - 5}

  • Domain: All real numbers except x=5x = 5 (can't divide by zero)

Finding Range from a Graph

Look at all the y-values the graph covers.

For a parabola opening upward:

  • Lowest point is the vertex
  • Range: [k,∞)[k, \infty) where kk is the y-coordinate of vertex

Interval Notation

  • (a,b)(a, b): open interval (doesn't include endpoints)
  • [a,b][a, b]: closed interval (includes endpoints)
  • (−∞,∞)(-\infty, \infty): all real numbers

📚 Practice Problems

1Problem 1easy

❓ Question:

Find the domain of f(x)=3x−7f(x) = 3x - 7

💡 Show Solution

This is a linear function. There are no restrictions (no division by zero, no square roots).

We can substitute any real number for xx.

Answer: Domain: all real numbers or (−∞,∞)(-\infty, \infty)

2Problem 2medium

❓ Question:

Find the domain of g(x)=1x+3g(x) = \frac{1}{x + 3}

💡 Show Solution

Look for restrictions. We cannot divide by zero.

Set the denominator equal to zero: x+3=0x + 3 = 0 x=−3x = -3

We must exclude x=−3x = -3 from the domain.

Answer: Domain: all real numbers except x=−3x = -3

In interval notation: (−∞,−3)∪(−3,∞)(-\infty, -3) \cup (-3, \infty)

3Problem 3hard

❓ Question:

Find the range of f(x)=x2−4x+1f(x) = x^2 - 4x + 1

💡 Show Solution

This is a quadratic function that opens upward (a=1>0a = 1 > 0).

Step 1: Find the vertex (the minimum point) x=−b2a=−−42(1)=2x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2

Step 2: Find the y-coordinate of the vertex f(2)=(2)2−4(2)+1=4−8+1=−3f(2) = (2)^2 - 4(2) + 1 = 4 - 8 + 1 = -3

The vertex is (2,−3)(2, -3).

Since the parabola opens upward, the minimum y-value is −3-3 and it goes to ∞\infty.

Answer: Range: [−3,∞)[-3, \infty)

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