Functions and Graphs
Functions and graphs questions in Bagrut ask you to connect an algebraic rule with its graph: domain, range, transformations, inverse, composition, asymptotes, and intervals of increase or decrease. A common task is to sketch a transformed function and identify its new key points.
Core rules
- For y=a·f(b(x−h))+k, horizontal changes act inside the argument and often feel reversed: x−h shifts right by h, b compresses horizontally by 1/b.
- Domain restrictions propagate: a denominator cannot be zero and an even root requires a nonnegative radicand.
- Inverse: swap x and y, solve for y, and state the inverse domain, which is the original range.
- f(g(x)) is defined only where x is in the domain of g and g(x) is in the domain of f.
- Vertical asymptotes come from excluded denominator values; horizontal asymptotes describe end behaviour.
Worked example
Let f(x)=√(3−x)+1. The radicand requires 3−x≥0, so domain x≤3. Since √(3−x)≥0, range y≥1. For the inverse, y−1=√(3−x). Square: (y−1)²=3−x, so x=3−(y−1)². Swap: f⁻¹(x)=3−(x−1)², with domain x≥1. The domain restriction is essential; without it the inverse is not a function on the full parabola.
Traps
Applying a horizontal shift in the wrong direction is the most common graph error. Squaring to find an inverse without recording the range creates an invalid branch. Students also forget that f∘g and g∘f are usually different. Do not read a graph's asymptote as a value in the domain.