1. **State the problem:** We are given the function $f(x) = 3x + 1$ defined on the domain $0 \leq x \leq 2$. We need to find the range of this function. 2. **Recall the formula and rules:** The range of a function is the set of all possible output values $f(x)$ when $x$ varies over the domain. 3. **Evaluate the function at the endpoints of the domain:** Since $f(x)$ is linear and continuous, the minimum and maximum values on the interval will occur at the endpoints. Calculate $f(0)$: $$f(0) = 3 \times 0 + 1 = 1$$ Calculate $f(2)$: $$f(2) = 3 \times 2 + 1 = 6 + 1 = 7$$ 4. **Determine the range:** Since $f(x)$ increases as $x$ increases (because the slope 3 is positive), the range is all values from $f(0) = 1$ to $f(2) = 7$. Therefore, the range is: $$[1, 7]$$ This means $f(x)$ takes all values between 1 and 7 inclusive when $x$ is between 0 and 2 inclusive.