Questions: Given (f(x)=2x+4) and (g(x)=5x^2) then find: a) ((f circ g)(x)=) b) ((g circ f)(x)=)

Transcript text: Given $f(x)=2 x+4$ and $g(x)=5 x^{2}$ then find: a) $(f \circ g)(x)=$ $\square$ b) $(g \circ f)(x)=$ $\square$

Solution

Solution Steps

To solve the given problem, we need to find the composition of two functions $ f(x) $ and $ g(x) $. The composition of functions involves substituting one function into another.

a) For $ (f \circ g)(x) $, we substitute $ g(x) $ into $ f(x) $. b) For $ (g \circ f)(x) $, we substitute $ f(x) $ into $ g(x) $.

Solution Approach

Define the functions $ f(x) $ and $ g(x) $.
Compute $ (f \circ g)(x) $ by substituting $ g(x) $ into $ f(x) $.
Compute $ (g \circ f)(x) $ by substituting $ f(x) $ into $ g(x) $.

Step 1: Define the Functions

We are given the functions: \[ f(x) = 2x + 4 \] \[ g(x) = 5x^2 \]

Step 2: Compute $ (f \circ g)(x) $

To find $ (f \circ g)(x) $, we substitute $ g(x) $ into $ f(x) $: \[ (f \circ g)(x) = f(g(x)) = f(5x^2) = 2(5x^2) + 4 = 10x^2 + 4 \] Evaluating this at $ x = 1 $: \[ (f \circ g)(1) = 10(1)^2 + 4 = 10 + 4 = 14 \]

Step 3: Compute $ (g \circ f)(x) $

To find $ (g \circ f)(x) $, we substitute $ f(x) $ into $ g(x) $: \[ (g \circ f)(x) = g(f(x)) = g(2x + 4) = 5(2x + 4)^2 \] Calculating $ (2x + 4)^2 $: \[ (2x + 4)^2 = 4x^2 + 16x + 16 \] Thus, \[ g(f(x)) = 5(4x^2 + 16x + 16) = 20x^2 + 80x + 80 \] Evaluating this at $ x = 1 $: \[ (g \circ f)(1) = 20(1)^2 + 80(1) + 80 = 20 + 80 + 80 = 180 \]

Final Answer

The results of the compositions are: \[ (f \circ g)(1) = 14 \] \[ (g \circ f)(1) = 180 \] Thus, the final answers are: \[ \boxed{(f \circ g)(1) = 14} \] \[ \boxed{(g \circ f)(1) = 180} \]

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