1. The problem asks to fill in the blanks about functions and answer questions about independent and dependent variables. 2. A function is a rule that assigns to each value of the **(1) independent variable** in the **(2) domain** a unique value of the **(3) dependent variable** in the **(4) range**. 3. The independent variable of a function belongs to the **domain**. The dependent variable belongs to the **range**. 4. Graph and analyze the functions: a. For $f(x) = 5x^4 - 13$ on domain $[-2, 2]$: - Calculate values at key points: $f(-2) = 5(-2)^4 - 13 = 5(16) - 13 = 80 - 13 = 67$ $f(0) = 5(0)^4 - 13 = -13$ $f(2) = 5(2)^4 - 13 = 5(16) - 13 = 67$ - Since $x^4$ is always non-negative and grows quickly, the minimum value is at $x=0$ with $f(0) = -13$. - The domain is $[-2, 2]$ as given. - The range is $[-13, 67]$ based on calculations. b. For $f(x) = \sqrt{x} - 1$ on domain $[-4, 4]$: - The square root function $\sqrt{x}$ is only defined for $x \geq 0$. - So the effective domain is $[0, 4]$. - Calculate values: $f(0) = \sqrt{0} - 1 = -1$ $f(4) = \sqrt{4} - 1 = 2 - 1 = 1$ - The range is $[-1, 1]$. - The given window $[-4,4] \times [-4,4]$ includes the domain and range. Final answers: - Fill in the blanks: (1) independent variable, (2) domain, (3) dependent variable, (4) range. - Independent variable belongs to the domain. - Dependent variable belongs to the range. - Domain and range for $f(x) = 5x^4 - 13$ are $[-2, 2]$ and $[-13, 67]$ respectively. - Domain and range for $f(x) = \sqrt{x} - 1$ are $[0, 4]$ and $[-1, 1]$ respectively.