Example 1.2 (a): f(x) = (x−4)2 + 5

Here are the domain and range for the function:

• Example 1.2 (a): $f(x) = (x-4)^2 + 5$

  • Step 1: Find the domain. This is a polynomial function (specifically, a quadratic function). Polynomials are defined for all real numbers. The domain is $D(f) = (-\infty, \infty)$ or $\mathbb{R}$.

  • Step 2: Find the range. The function is in the vertex form $f(x) = a(x-h)^2 + k$, where the vertex is $(h, k)$. In this case, $a=1$, $h=4$, and $k=5$. Since $a=1 > 0$, the parabola opens upwards, meaning the vertex is the minimum point of the function. The minimum value of $f(x)$ is $k=5$. Therefore, $f(x) \ge 5$. The range is $R(f) = [5, \infty)$.

  • Final Answer for (a): Domain: $\boxed{(-\infty, \infty)}$ Range: $\boxed{[5, \infty)}$