Suppose \( f(x)=\sqrt{x^{2}+4 x+5} \) and \( g(x)=-9 x-5 \) \( (f \circ g)(x)=\square \) \( (f \circ g)(3)=\square \)

To find \( (f \circ g)(x) \), we need to evaluate \( f(g(x)) \). First, let's calculate \( g(x) \): \[ g(x) = -9x - 5 \] Now, we can substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(-9x - 5) = \sqrt{(-9x - 5)^2 + 4(-9x - 5) + 5} \] Now let's simplify it step by step: 1. Compute \( (-9x - 5)^2 \): \[ (-9x - 5)^2 = 81x^2 + 90x + 25 \] 2. Compute \( 4(-9x - 5) \): \[ 4(-9x - 5) = -36x - 20 \] 3. Combine all the parts: \[ (-9x - 5)^2 + 4(-9x - 5) + 5 = (81x^2 + 90x + 25) + (-36x - 20) + 5 \] \[ = 81x^2 + (90x - 36x) + (25 - 20 + 5) = 81x^2 + 54x + 10 \] Thus, we have: \[ (f \circ g)(x) = f(g(x)) = \sqrt{81x^2 + 54x + 10} \] Next, let's calculate \( (f \circ g)(3) \): \[ g(3) = -9(3) - 5 = -27 - 5 = -32 \] Now plugging in: \[ (f \circ g)(3) = f(-32) = \sqrt{(-32)^2 + 4(-32) + 5} \] Calculating further: \[ (-32)^2 = 1024, \quad 4(-32) = -128 \] So, \[ f(-32) = \sqrt{1024 - 128 + 5} = \sqrt{1024 - 128 + 5} = \sqrt{901} \] In conclusion: \[ (f \circ g)(x) = \sqrt{81x^2 + 54x + 10} \] \[ (f \circ g)(3) = \sqrt{901} \]