If () = − 1 and () =  2 + − 6, for what values of is the  expression ∘ () equal to zero?

The values of \( x \) that make \( (f \circ g)(x) = 0 \) are:\[x = \frac{-1 + \sqrt{29}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{29}}{2}\]Step-by-step explanation:Let’s break this down step by step. You’re given:- \( f(x) = x - 1 \) - \( g(x) = x^2 + x - 6 \) - You’re asked to find the values of \( x \) for which \( (f \circ g)(x) = 0 \) Step 1: Understand the compositionThe notation \( (f \circ g)(x) \) means:> Apply \( g(x) \) first, then plug that result into \( f \)So:\[(f \circ g)(x) = f(g(x)) = g(x) - 1\] Step 2: Plug in \( g(x) \)\[f(g(x)) = (x^2 + x - 6) - 1 = x^2 + x - 7\] Step 3: Solve for when this equals zero\[x^2 + x - 7 = 0\]This is a quadratic equation. Let’s solve it using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Where:- \( a = 1 \)- \( b = 1 \)- \( c = -7 \)\[x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-7)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 28}}{2} = \frac{-1 \pm \sqrt{29}}{2}\]✅

Answered by kylemolo1212 | 2025-08-26