Questions: Consider the function f(x) = sqrt(-7x+8) - 1. What is the domain and range of this function? Domain x ≥ -1 and range f(x) ≥ -8 Domain x ≥ 8 / 7 and range f(x) ≤ -1 Domain x ≤ -8 and range f(x) ≥ -1 Domain x ≤ 8 / 7 and range f(x) ≥ -1 Domain x ≤ -8 / 7 and range f(x) ≥ -1

Solution Steps

To determine the domain of the function \( f(x) = \sqrt{-7x + 8} - 1 \), we need to ensure that the expression inside the square root is non-negative, i.e., \(-7x + 8 \geq 0\). Solving this inequality will give us the domain. For the range, we consider the minimum value of the square root function, which is 0, and adjust for the subtraction of 1.

To find the domain of the function \( f(x) = \sqrt{-7x + 8} - 1 \), we need to ensure that the expression inside the square root is non-negative. This means solving the inequality:

\[ -7x + 8 \geq 0 \]

Solving for \( x \), we get:

\[ -7x \geq -8 \quad \Rightarrow \quad x \leq \frac{8}{7} \]

Thus, the domain of the function is \( x \leq \frac{8}{7} \).

The range of the function is determined by the values that \( f(x) \) can take. The minimum value of the square root function \(\sqrt{-7x + 8}\) is 0, which occurs when \( x = \frac{8}{7} \). Therefore, the minimum value of \( f(x) \) is:

\[ f(x) = \sqrt{0} - 1 = -1 \]

Since the square root function can take any non-negative value, \( f(x) \) can take any value greater than or equal to \(-1\). Thus, the range of the function is \( f(x) \geq -1 \).

Final Answer