Function Families and Parent Functions

Easy

1. Question: To which function family does  $f(x) = x$  belong? What is its corresponding parent function? Answer: It belongs to the linear function family, and its parent function is itself ( $f(x) = x$ ).
2. Question: Write the expression of the parent function for the quadratic function family and state the coordinates of its graph’s vertex. Answer: The parent function is  $f(x) = x^2$ , and the vertex coordinates are  $(0, 0)$ .

Medium

1. Question: Analyze the domain, range, and the intersection with the x-axis of the parent function  $f(x) = |x|$ . Answer:
   • Domain:  $\mathbb{R}$  (all real numbers);
   • Range:  $[0, +\infty)$  (non-negative real numbers);
   • Intersection with the x-axis:  $(0, 0)$  (at the vertex).
2. Question: Why does the domain of  $f(x) = \frac{1}{x}$  not include 0? What is the trend of its graph as x approaches 0? Answer:
   • The domain excludes 0 because the denominator cannot be zero;
   • As  $x \to 0^+$ ,  $f(x) \to +\infty$ ; as  $x \to 0^-$ ,  $f(x) \to -\infty$ .

Hard

1. Question: Compare the symmetry, domain, and monotonicity of the cubic parent function  $f(x) = x^3$  and the reciprocal parent function  $f(x) = \frac{1}{x}$ . Answer:
   • Symmetry: Both are symmetric about the origin (odd functions);
   • Domain:  $f(x) = x^3$  has a domain of  $\mathbb{R}$ , while  $f(x) = \frac{1}{x}$  has a domain of  $x \neq 0$ ;
   • Monotonicity:  $f(x) = x^3$  is monotonically increasing over its entire domain;  $f(x) = \frac{1}{x}$  is monotonically decreasing on  $(-\infty, 0)$  and  $(0, +\infty)$  separately but not monotonic over its entire domain.
2. Question: A function family has a parent function whose graph is a parabola. For this function, when $x > 0$ , the function value increases as x increases; when  $x < 0$ , the function value decreases as x increases. What is this function family? What is the expression of its parent function? Answer: The function family is the quadratic function family, and its parent function is  $f(x) = x^2$  (an upward-opening parabola with a minimum at  $x = 0$ , decreasing on the left and increasing on the right of the vertex).