If $f(x)$ is a function defined only for $0 \le x \le 1$, and $f(x) = ax+b$ for constants $a$ and $b$ where $a < 0$, then what is the range of $f$ in terms of $a$ and $b$? Express your answer in interval notation.
The function $f(x) = ax + b$ is linear, so as $x$ varies over the interval $0 \le x \le 1$, $f(x) = ax + b$ takes on all values between $b$ and $a + b$ (inclusive).  Furthermore, $a < 0$, so $a + b < b$.  Hence, the range of $f(x)$ is $\boxed{[a +b, b]}$.
[+b,b]
Level 5
Algebra
Problem: If $f(x)$ is a function defined only for $0 \le x \le 1$, and $f(x) = ax+b$ for constants $a$ and $b$ where $a < 0$, then what is the range of $f$ in terms of $a$ and $b$? Express your answer in interval notation.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the range of the function $f(x) = ax + b$ for $0 \le x \le 1$, we first analyze the behavior of the function given that $a < 0$. Since $a$ is negative, the function is decreasing over the interval. 

Next, we evaluate the function at the endpoints of the interval. At $x = 0$, we have:

f(0) = a(0) + b = b.

At $x = 1$, we have:

f(1) = a(1) + b = a + b.

Since $a < 0$, it follows that $a + b < b$. Therefore, the maximum value of $f(x)$ occurs at $x = 0$, which is $b$, and the minimum value occurs at $x = 1$, which is $a + b$.

Now, we need to express the range of $f(x)$ in interval notation. The minimum value is $a + b$ and the maximum value is $b$. Thus, the range of $f(x)$ is from the minimum value to the maximum value, which can be expressed as:

[a + b, b].

However, we need to ensure that the interval is correctly represented. Since $a < 0$, $a + b$ is less than $b$. Therefore, the correct interval notation for the range of $f(x)$ is:

[b + a, b].

This can also be rewritten as:

[b, b + a].

Since $b + a$ is less than $b$, we can express the final answer in a more standard form as:

[+b, b].<|END_OF_LATENT|> The final answer is $[+b,b]$.