1. **Determine discontinuities of** $f(x) = \frac{x^2 - 3x - 10}{x + 2}$. - Factor numerator: $x^2 - 3x - 10 = (x - 5)(x + 2)$. - Simplify: $f(x) = \frac{(x - 5)(x + 2)}{x + 2}$, for $x \neq -2$. - Discontinuity at $x = -2$ because denominator is zero. - Since $(x+2)$ cancels, this is a removable discontinuity (hole). 2. **Determine discontinuities of piecewise function** $$f(x) = \begin{cases} x + 3 & x \geq 2 \\ x^2 + 1 & x < 2 \end{cases}$$ - Check continuity at $x=2$. - Left limit: $\lim_{x \to 2^-} x^2 + 1 = 4 + 1 = 5$. - Right limit: $\lim_{x \to 2^+} x + 3 = 2 + 3 = 5$. - Function value: $f(2) = 2 + 3 = 5$. - Limits and function value equal, so continuous at $x=2$. - No discontinuities. 3. **Determine discontinuities of** $f(x) = \frac{x^4 - 1}{x^2 - 1}$. - Factor numerator: $x^4 - 1 = (x^2 - 1)(x^2 + 1)$. - Denominator: $x^2 - 1 = (x - 1)(x + 1)$. - Simplify: $f(x) = \frac{(x^2 - 1)(x^2 + 1)}{x^2 - 1} = x^2 + 1$, for $x \neq \pm 1$. - Discontinuities at $x = 1$ and $x = -1$ (denominator zero). - Since $(x^2 - 1)$ cancels, these are removable discontinuities. 4. **Determine discontinuities of** $f(x) = \frac{x^3 + x^2 - 17x + 15}{x^2 + 2x - 15}$. - Factor denominator: $x^2 + 2x - 15 = (x + 5)(x - 3)$. - Find zeros of numerator by testing roots: - $f(3) = 27 + 9 - 51 + 15 = 0$, so $(x - 3)$ is factor. - Divide numerator by $(x - 3)$: quotient $x^2 + 4x - 5$. - Factor quotient: $(x + 5)(x - 1)$. - Numerator factors: $(x - 3)(x + 5)(x - 1)$. - Simplify $f(x) = \frac{(x - 3)(x + 5)(x - 1)}{(x + 5)(x - 3)} = x - 1$, for $x \neq -5, 3$. - Discontinuities at $x = -5$ and $x = 3$. - Both are removable discontinuities (factors cancel). 5. **Determine discontinuities of polynomial** $f(x) = x^3 - 7x$. - Polynomials are continuous everywhere. - No discontinuities. 6. **Determine discontinuities of** $f(x) = \frac{x^2 - 4}{x^2 - 5x + 6}$. - Factor numerator: $x^2 - 4 = (x - 2)(x + 2)$. - Factor denominator: $x^2 - 5x + 6 = (x - 2)(x - 3)$. - Simplify: $f(x) = \frac{(x - 2)(x + 2)}{(x - 2)(x - 3)} = \frac{x + 2}{x - 3}$, for $x \neq 2$. - Discontinuities at $x = 2$ and $x = 3$. - At $x=2$, removable discontinuity (factor cancels). - At $x=3$, vertical asymptote (denominator zero, no cancellation). 7. **Determine discontinuities of** $f(x) = \frac{x^2 + 3x + 2}{x^2 + 4x + 3}$. - Factor numerator: $(x + 1)(x + 2)$. - Factor denominator: $(x + 1)(x + 3)$. - Simplify: $f(x) = \frac{(x + 1)(x + 2)}{(x + 1)(x + 3)} = \frac{x + 2}{x + 3}$, for $x \neq -1$. - Discontinuities at $x = -1$ and $x = -3$. - At $x = -1$, removable discontinuity. - At $x = -3$, vertical asymptote. **Summary:** - Discontinuities occur where denominator is zero. - If factor cancels with numerator, discontinuity is removable (hole). - If factor does not cancel, discontinuity is infinite (vertical asymptote). Check graphs on graphing calculators to confirm these discontinuities visually.