Exercises

1. Given $f(x) = x^3 - 10x^2 + 31x - 30$; show that
   $$f(0) = -30,\ \ \ \ f(y) = y^3 - 10y^2 + 31y - 30,$$
   $$f(2) = 0,\ \ \ \ f(a) = a^3 - 10a^2 + 31a - 30,$$
   $$f(3) = f(5),\ \ \ \ f(yz) = y^3z^3 - 10y^2z^2 + 31yz - 30,$$
   $$f(1) > f( − 3),\ \ \ \ f(x − 2) = x^3 - 16x^2 + 83x - 140,$$
   $$f( - 1) = 6f(6).$$

2. If $f(x) = x^3 - 3x + 2$, find $f(0)$, $f(1)$, $f(-1)$, $f \left ( -\frac{1}{2} \right )$, $f \left ( \frac{4}{3} \right )$.

3. If $f(x) = x^3 - 10x^2 + 31x - 30$, and $\phi (x) = x^4 − 55x^2 − 210x − 216$, show that

   $f(2) = \phi ( - 2)$, $f(3) = \phi( - 3),f(5) = \phi( - 4)$, $f(0) + \phi (0) + 246 = 0$.

4. If $F(x) = 2x$, find $F(0)$, $F(-3)$, $F \left ( \frac{1}{3} \right )$, $F(-1)$.

5. Given $F(x) = x(x - 1)(x + 6) \left ( x - \frac{1}{2} \right ) \left (x + \frac{5}{4} \right )$, show that $F(0) = F(1) = F(-6) = F \left (\frac{1}{2} \right ) = F \left ( -\frac{5}{4} \right ) = 0$.

6. If $f(m_1) = \frac{m_1 - 1}{m_1 + 1}$, show that $\frac{f(m_1) - f(m_2)}{1 + f(m_1)f(m_2)} = \frac{m_1 - m_2}{1 + m_1 m_2}$.

7. If $\phi (x) = a^x$, show that $\phi(y) \cdot \phi(z) = \phi(y + z)$.

8. Given $\phi(x) = \log \frac{1 - x}{1 + x}$, show that $\phi(x) + \phi(y) = \phi \left ( \frac{x + y}{1 + xy} \right )$.

9. If $f(\phi ) = \cos\phi$, show that $f(\phi ) = f( - \phi ) = - f(\pi- \phi) = - f(\pi + \phi)$.

10. If $F(\theta) = \tan\theta$, show that $F(2\theta) = \frac{2F(\theta)}{1 - [ F(\theta) ]^2}$.

11. Given $\psi(x) = x^{2n} + x^{2m} + 1$, show that $\psi(1) = 3$, $\psi(0) = 1$, and $\psi(a) = \psi(-a)$.

12. If $f(x) = \frac{2x - 3}{x + 7}$, find $f(\sqrt{2})$.

Here's how to verify the double angle identity for $\tan$ in Exercise 10 above:

[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\sage]

sage: theta = var("theta")
sage: tan(2*theta).expand_trig()
2*tan(theta)/(1 - tan(theta)^2)