Example and solutions

• Let $f(x)=x^2-x-6$.

  • Does this actually define a function?
    
    Yes. For each input $x$, there is exactly one output value. That value is the single result of the computations $x^2-x-6$.
    
    

  • What is the domain of $f$?
    
    The expression $x^2-x-6$ is defined for any replacement of the variable $x$. The domain is the set of all real numbers: $(-\infty,\infty)$.
    
    

  • Evaluate $f(1)$.
    
    $f(1)=(1)^2-(1)-6=1-1-6=-6$
    
    

  • Find the $x$- and $y$-intercepts of the graph of $f$.
    
    $x$-intercepts: Set $y=0$ to get $0=x^2-x-6=(x-3)(x+2)$. The $x$-intercepts are $(3,0)$ and $(-2,0)$.
    $y$-intercept: Set $x=0$ to get $y=f(0)=-6$. The $y$-intercept is $(0,-6)$.
    
    

  • Evaluate $f(t)$.
    
    $f(t)=(t)^2-(t)-6=t^2-t-6$
    
    

  • Evaluate and expand $f(t-2)$.
    

    $$f(t-2)=(t-2)^2-(t-2)-6$$
    $$=t^2-4t+4-t+2-6$$
    $$=t^2-5t$$