1. The error in the function \( f(x) = \frac{2x + 3}{x - 4} \) is that it is undefined at \( x = 4 \), because the denominator becomes zero, which is not allowed in division.
2. The error in the function \( f(x) = \sqrt{x^2 + 1} \) is that there is no error; the expression under the square root, \( x^2 + 1 \), is always positive for all real numbers \( x \), so the function is defined for all real numbers.
3. The domain restrictions are:
- \( f(x) = \sqrt{x - 5} \): \( x \geq 5 \)
- \( f(x) = \frac{1}{x - 2} \): \( x \neq 2 \)
- \( f(x) = \sqrt{4 - x} \): \( x \leq 4 \)
- \( f(x) = \frac{1}{\sqrt{x}} \): \( x > 0 \)
4. For the function \( f(x) = \sqrt{2x - 6} \), the domain restriction is \( x \geq 3 \). This is because the expression inside the square root, \( 2x - 6 \), must be greater than or equal to zero: \( 2x - 6 \geq 0 \) → \( 2x \geq 6 \) → \( x \geq 3 \).
Parent Tip: Review the logic above to help your child master the concept of algebra with pizzazz 155 answer.
