Math Problem Statement

If f(x) = 26 + 3

x + 13 and g(x) = 2x2 + 5, compute the composition (f ∘ g)(3).

(f ∘ g)(3)

Solution

To compute the composition (fg)(3)(f \circ g)(3), we need to evaluate g(3)g(3) first and then use that result as the input for f(x)f(x).

Step 1: Compute g(3)g(3)

Given g(x)=2x2+5g(x) = 2x^2 + 5, we can substitute x=3x = 3:

g(3)=2(3)2+5g(3) = 2(3)^2 + 5

Calculate 323^2:

32=93^2 = 9

So,

g(3)=2×9+5=18+5=23g(3) = 2 \times 9 + 5 = 18 + 5 = 23

Step 2: Compute f(g(3))=f(23)f(g(3)) = f(23)

Given f(x)=26+3x+13f(x) = \frac{26 + 3}{x + 13}, substitute x=23x = 23:

f(23)=26+323+13f(23) = \frac{26 + 3}{23 + 13}

Simplify the expression:

f(23)=2936f(23) = \frac{29}{36}

Conclusion

Thus, the value of the composition (fg)(3)(f \circ g)(3) is:

(fg)(3)=2936(f \circ g)(3) = \frac{29}{36}

Would you like further details or have any questions?

Here are some related questions you might be interested in:

  1. How do you find the composition of two functions in general?
  2. What is the domain of the composite function f(g(x))f(g(x))?
  3. How does the composition of functions relate to real-world applications?
  4. Can a composite function ever be undefined?
  5. What are some properties of composite functions?

Tip: When working with compositions of functions, always evaluate from the innermost function outward.

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Math Problem Analysis

Mathematical Concepts

Composition of functions
Quadratic equations
Fractional expressions

Formulas

Composition of functions formula
Quadratic equation formula

Theorems

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Suitable Grade Level

High school