How Do You Find Parameters to Make a Piecewise Function Continuous?

You find the parameters by making the formulas on both sides of every boundary point produce the same output. This gives one equation per join; solve these equations simultaneously for the unknown parameters.

This is AHL 2.9: HL modelling functions, so it is HL-only content and may appear in Paper 1 or Paper 2. A function is continuous at $x=c$ when

$$\lim_{x\to c^-}f(x)=f(c)=\lim_{x\to c^+}f(x).$$

The formal definition of continuity is not required, but it explains the method. If each individual formula is continuous within its own interval, only the joins must be checked.

Consider

$$f(x)=\begin{cases} x+2, & x<1,\\ ax+b, & 1\leq x<3,\\ 2x-1, & x\geq 3. \end{cases}$$

There are two boundaries, so two matching equations are needed.

Now solve the simultaneous equations:

$$\begin{aligned} a+b&=3,\\ 3a+b&=5. \end{aligned}$$

Each equation represents a necessary condition for a join. With two unknown parameters, the two independent boundary conditions normally determine a unique pair. If there were three unknowns but only two joins, continuity alone would usually leave one free parameter. Conversely, inconsistent boundary equations would show that no values can make every join continuous. This equation-counting check helps you decide whether the available information is sufficient before using technology.

Subtracting the first equation from the second gives

$$2a=2 \quad\Rightarrow\quad a=1.$$

Substituting $a=1$ into $a+b=3$ gives $b=2$. Therefore, the required parameters are $a=1$ and . Checking confirms that both adjacent formulas give at and at .

A common misconception is to evaluate only the piece whose interval contains the equality sign. This is incorrect because continuity depends on both the left-hand limit and right-hand limit, regardless of which piece defines the function at the boundary.

Exam technique: write a matching equation for every boundary before solving with your GDC. Examiners need to see the continuity conditions; unexplained parameter values may not receive the method marks. Keep full calculator precision if the equations produce non-integer values, and give exact answers unless the question requests an approximation.