IB Mathematics Applications & Interpretation Question Type 8: Determining Continuity of Piecewise Models Exercises

Express $b$ in terms of $a$ so that the function $$f(x)=\{\begin{array}{ll}{x}^2,& x\le 2\\ ax+b,& x>2\end{array}$$ is continuous at $x=2$.

Determine $a$ so that $$f(x)=\{\begin{array}{ll}\sqrt{x+3},& -3\le x<1,\\ a{x}^2+1,& x\ge 1\end{array}$$ is continuous at $x=1$.

Determine $b$ so that $$f(x)=\{\begin{array}{ll}3x+b,& x<2,\\ 5x+1,& x\ge 2\end{array}$$ is continuous at $x=2$.

Find $k$ so that $$f(x)=\{\begin{array}{ll}\frac{\sin x}{x},& x\ne 0,\\ k,& x=0\end{array}$$ is continuous at $x=0$.

Find the value of $a$ such that the function $$f(x)=\{\begin{array}{ll}{x}^2+ax+1,& x\le 1\\ 2ax+1,& x>1\end{array}$$ is continuous at $x=1$.

Find $a$ so that $$f(x)=\{\begin{array}{ll}{x}^3-8,& x<2,\\ a(x-2),& x\ge 2\end{array}$$ is continuous at $x=2$.

Find $a$ so that the function $$f(x)=\{\begin{array}{ll}\frac{x^2-1}{x-1},& x\ne 1\\ a,& x=1\end{array}$$ is continuous at $x=1$.

Find the value of $a$ such that the function $$f(x)=\{\begin{array}{ll}x+2,& x<1\\ ax-3,& x\ge 1\end{array}$$ is continuous at $x=1$.

Given the piecewise function $$f(x)=\{\begin{array}{ll}ax+b,& x<1,\\ 2x+1,& 1\le x<3,\\ cx-1,& x\ge 3,\end{array}$$ find $b$ and $c$ in terms of $a$ so that $f$ is continuous at both $x=1$ and $x=3$.

Determine $m$ so that $$f(x)=\{\begin{array}{ll}(x-1{)}^2,& x<1\\ mx+1,& x\ge 1\end{array}$$ is continuous at $x=1$.

Determine the value of $a$ so that $$f(x)=\{\begin{array}{ll}x+2,& x<0\\ a{x}^2+2,& 0\le x<1\\ 2x+1,& x\ge 1\end{array}$$ is continuous on its domain.

Determine $k$ such that $$f(x)=\{\begin{array}{ll}kx+3,& x<3\\ 2x-1,& x\ge 3\end{array}$$ is continuous at $x=3$.