Question
Suppose \( f(x)=7 x+9 \) and \( g(x)=\sqrt{x^{2}-3 x+6} \) \( (f \circ g)(x)=\square \) \( (f \circ g)(3)=\square \)
Ask by Delgado Floyd. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( (f \circ g)(x) = 7\sqrt{x^2 - 3x + 6} + 9 \)
\( (f \circ g)(3) \approx 26.146428 \)
Solution
To find the composition of functions \( f \circ g \), we need to substitute the expression for \( g(x) \) into the function \( f(x) \).
Given:
\( f(x) = 7x + 9 \)
\( g(x) = \sqrt{x^2 - 3x + 6} \)
Substitute \( g(x) \) into \( f(x) \):
\( (f \circ g)(x) = f(g(x)) = 7(g(x)) + 9 \)
Substitute the expression for \( g(x) \) into the equation:
\( (f \circ g)(x) = 7\sqrt{x^2 - 3x + 6} + 9 \)
Now, we need to find \( (f \circ g)(3) \) by substituting \( x = 3 \) into the expression for \( (f \circ g)(x) \):
\( (f \circ g)(3) = 7\sqrt{3^2 - 3(3) + 6} + 9 \)
Let's calculate \( (f \circ g)(3) \) step by step.
The composition of functions \( (f \circ g)(x) = 7\sqrt{x^2 - 3x + 6} + 9 \).
To find \( (f \circ g)(3) \), we substitute \( x = 3 \) into the expression:
\( (f \circ g)(3) = 7\sqrt{3^2 - 3(3) + 6} + 9 \)
Calculating \( (f \circ g)(3) \):
\( (f \circ g)(3) = 7\sqrt{6} + 9 \)
Therefore, \( (f \circ g)(3) \approx 26.146428 \).
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
To find \( (f \circ g)(x) \), we first need to compute \( g(x) \) and then substitute that into \( f(x) \). Given \( g(x) = \sqrt{x^2 - 3x + 6} \), we first evaluate \( g(3) \): \[ g(3) = \sqrt{3^2 - 3(3) + 6} = \sqrt{9 - 9 + 6} = \sqrt{6} \] Now, we can find \( (f \circ g)(x) = f(g(x)) \): \[ f(g(x)) = f\left(\sqrt{x^2 - 3x + 6}\right) = 7\left(\sqrt{x^2 - 3x + 6}\right) + 9 \] Thus, \[ (f \circ g)(x) = 7\sqrt{x^2 - 3x + 6} + 9 \] Now, substituting \( x = 3 \) into \( (f \circ g)(x) \): \[ (f \circ g)(3) = 7\sqrt{3^2 - 3(3) + 6} + 9 = 7\sqrt{6} + 9 \] So the solutions are: \[ (f \circ g)(x) = 7\sqrt{x^2 - 3x + 6} + 9 \] \[ (f \circ g)(3) = 7\sqrt{6} + 9 \]
