Determine the domain and range of the following algebraic expression X-1/ x²+3x-18

To determine the domain and range of the algebraic expression, we need to consider any restrictions on the variables that would make the expression undefined or affect its output values. The expression is: f(x) = (x - 1) / (x^2 + 3x - 18) 1. Domain: The expression is a rational function, and the denominator (x^2 + 3x - 18) should not equal zero since division by zero is undefined. To find the values that make the denominator zero, we can factorize it: x^2 + 3x - 18 = (x + 6)(x - 3) Setting each factor equal to zero, we have: x + 6 = 0 --> x = -6 x - 3 = 0 --> x = 3 Therefore, the values -6 and 3 make the denominator zero. Hence, the domain of the expression is all real numbers except -6 and 3. In interval notation, the domain can be expressed as (-∞, -6) U (-6, 3) U (3, ∞). 2. Range: To determine the range, we need to consider the behavior of the expression as x approaches positive or negative infinity. As x approaches positive or negative infinity, the expression approaches either positive infinity or negative infinity, depending on the sign of the numerator and denominator. For this expression, as x approaches positive or negative infinity: - If the numerator (x - 1) is positive, and the denominator (x^2 + 3x - 18) is positive, the expression approaches positive infinity. - If the numerator (x - 1) is negative, and the denominator (x^2 + 3x - 18) is positive, the expression approaches negative infinity. - If the denominator (x^2 + 3x - 18) approaches zero (at x = -6 or x = 3), the expression is undefined. Therefore, the range of the expression is (-∞, ∞)