Question

for the following exercise, find the domain of the function using interval notation.
$f(x) = \frac{4x + 5}{\sqrt{5 - x}}$

Explanation:

Step1: Analyze the denominator

The function is a fraction with a square root in the denominator: \( f(x)=\frac{4x + 5}{\sqrt{5 - x}} \). For the square root to be defined, the expression inside (the radicand) must be non - negative. But since it's in the denominator, it can't be zero (because division by zero is undefined). So we need \( 5 - x>0 \).

Step2: Solve the inequality

Solve the inequality \( 5 - x>0 \). Subtract 5 from both sides: \( -x>- 5 \). Multiply both sides by - 1 (and remember to reverse the inequality sign when multiplying or dividing by a negative number), we get \( x < 5 \).

Answer:

\((-\infty, 5)\)