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Day 1 · Learn3 min read
Functions: Notation, Domain & Range
A function is a rule that takes one input and gives back exactly one output. We usually write the rule with function notation, like f(x) = 2x + 1 Here x is the input and f(x) (read "f of x") is the output. To evaluate, replace every x with the number you're given.
Example. For f(x) = 2x + 1, find f(3). Substitute: f(3) = 2(3) + 1 = 6 + 1 = 7. So the input 3 pairs with the output 7.
Domain and range
The domain is the set of all allowed inputs (the x values). The range is the set of all outputs (the f(x) values). Most functions accept every real number, but two situations shrink the domain:
You cannot divide by zero — so any x that makes a denominator 0 is excluded.
You cannot take a square root of a negative number — so keep whatever is under a √ at ≥ 0.
Finding a restricted domain
Look at the expression, set up the safety condition, and solve for x. For √(x − 4), you need x − 4 ≥ 0, which gives x ≥ 4. For 1/(x − 5), you need x − 5 ≠ 0, so x ≠ 5. Everything else stays in the domain. Range often comes from reading the graph or the lowest/highest output the rule can produce.
Now work today's 5 problems ↓
Q1
If f(x) = 4x − 9, what is f(−3)?
Substitute x = −3 into the rule: 4(−3) − 9. First 4 × (−3) = −12, then −12 − 9 = −21. Keeping the negative sign on the product and still subtracting 9 gives −21.
Q2
What is the domain of g(x) = √(x − 4)?
A square root is defined only when its inside is not negative, so x − 4 ≥ 0. Adding 4 to both sides gives x ≥ 4. Equality is allowed because √0 = 0 is defined.
Q3
The graph shows a parabola f(x) with its lowest point at (−1, −4). What is the range of f?
Range is the set of output (y) values. The parabola opens upward, so its vertex is the minimum output. The vertex output is the y-value −4, and every point above it is reached, giving y ≥ −4. The −1 is the x-coordinate of the vertex, not an output.
Q4
For f(x) = 2x + 3, find the value of x for which f(x) = 19.
Set the rule equal to 19: 2x + 3 = 19. Subtract 3 from both sides to get 2x = 16, then divide by 2 to get x = 8. Both undoing steps must be applied in order.
Q5
What is the domain of h(x) = √(x + 5) / (x − 2)?
Two conditions apply. The square root requires x + 5 ≥ 0, so x ≥ −5 (equality allowed since √0 is defined). The denominator cannot be zero, so x − 2 ≠ 0, meaning x ≠ 2. Combining both gives x ≥ −5 with x ≠ 2.
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