Let's go through each part of this problem.

Problem Statement:

We are asked to:

1. Determine the symmetry of each function.
2. Identify the domain and range in three different formats:
   • Inequality notation.
   • Interval notation.
   • Line notation on the x-axis.

Solutions

Part (a): $f(x) = x^4 - 13x + 36$

1. Symmetry:

   • This function has no symmetry.
   • Polynomial functions with both odd and even powers generally do not exhibit symmetry with respect to the x-axis, y-axis, or origin.

2. Domain:

   • Since this is a polynomial function, the domain is all real numbers.
   • Interval Notation: $(-\infty, \infty)$
   • Inequality Notation: $x \in \mathbb{R}$

3. Range:

   • Since $f(x) = x^4 - 13x + 36$ is a quartic function, it tends to infinity as $x$ approaches infinity. The minimum value can be determined by analyzing the critical points.
   • Range: $[y_{\text{min}}, \infty)$ where $y_{\text{min}}$ is the minimum value found from analysis or graphing.

Part (b): $f(x) = x^2$

1. Symmetry:

   • The function $f(x) = x^2$ is symmetric with respect to the y-axis (even function).

2. Domain:

   • The domain is all real numbers.
   • Interval Notation: $(-\infty, \infty)$
   • Inequality Notation: $x \in \mathbb{R}$

3. Range:

   • Since $x^2 \geq 0$, the range is all non-negative real numbers.
   • Interval Notation: $[0, \infty)$
   • Inequality Notation: $f(x) \geq 0$

Part (c): $f(x) = x$

1. Symmetry:

   • The function $f(x) = x$ has origin symmetry (odd function).

2. Domain:

   • The domain is all real numbers.
   • Interval Notation: $(-\infty, \infty)$
   • Inequality Notation: $x \in \mathbb{R}$

3. Range:

   • The range is all real numbers.
   • Interval Notation: $(-\infty, \infty)$
   • Inequality Notation: $f(x) \in \mathbb{R}$

Part (d): $f(x) = (x - 3)^3$

1. Symmetry:

   • This function has origin symmetry (odd function).

2. Domain:

   • The domain is all real numbers.
   • Interval Notation: $(-\infty, \infty)$
   • Inequality Notation: $x \in \mathbb{R}$

3. Range:

   • The range is all real numbers since the cubic function covers all $y$-values.
   • Interval Notation: $(-\infty, \infty)$
   • Inequality Notation: $f(x) \in \mathbb{R}$

Would you like any further details on a specific part of this problem?

Related Questions

1. How would the range of a function change if it had a minimum or maximum point?
2. Why does a polynomial function of odd degree have an unrestricted range?
3. How do you determine if a function is even, odd, or neither?
4. How can symmetry help in sketching graphs of functions quickly?
5. How does shifting a function (e.g., $(x-3)^3$) affect its symmetry?

Tip:

For polynomials, check the degree and leading term for insights into the function's end behavior and potential symmetry.