Question

given f(x) = 7x and g(x) = 2x² + 6, find the following expressions.
(a) (f ∘ g)(4) (b) (g ∘ f)(2) (c) (f ∘ f)(1) (d) (g ∘ g)(0)

(a) (f ∘ g)(4) = \square (simplify your answer.)
(b) (g ∘ f)(2) = \square (simplify your answer.)
(c) (f ∘ f)(1) = \square (simplify your answer.)
(d) (g ∘ g)(0) = \square (simplify your answer.)

Explanation:

Part (a): Find \((f \circ g)(4)\)

Step 1: Recall the definition of function composition

The composition \((f \circ g)(x)\) means \(f(g(x))\). So first, we need to find \(g(4)\), and then substitute that result into \(f(x)\).

Step 2: Calculate \(g(4)\)

Given \(g(x) = 2x^2 + 6\), substitute \(x = 4\):

$$ g(4) = 2(4)^2 + 6 = 2(16) + 6 = 32 + 6 = 38 $$

Step 3: Calculate \(f(g(4)) = f(38)\)

Given \(f(x) = 7x\), substitute \(x = 38\):

$$ f(38) = 7(38) = 266 $$

Part (b): Find \((g \circ f)(2)\)

Step 1: Recall the definition of function composition

The composition \((g \circ f)(x)\) means \(g(f(x))\). So first, we need to find \(f(2)\), and then substitute that result into \(g(x)\).

Step 2: Calculate \(f(2)\)

Given \(f(x) = 7x\), substitute \(x = 2\):

$$ f(2) = 7(2) = 14 $$

Step 3: Calculate \(g(f(2)) = g(14)\)

Given \(g(x) = 2x^2 + 6\), substitute \(x = 14\):

$$ g(14) = 2(14)^2 + 6 = 2(196) + 6 = 392 + 6 = 398 $$

Part (c): Find \((f \circ f)(1)\)

Step 1: Recall the definition of function composition

The composition \((f \circ f)(x)\) means \(f(f(x))\). So first, we need to find \(f(1)\), and then substitute that result into \(f(x)\).

Step 2: Calculate \(f(1)\)

Given \(f(x) = 7x\), substitute \(x = 1\):

$$ f(1) = 7(1) = 7 $$

Step 3: Calculate \(f(f(1)) = f(7)\)

Given \(f(x) = 7x\), substitute \(x = 7\):

$$ f(7) = 7(7) = 49 $$

Part (d): Find \((g \circ g)(0)\)

Step 1: Recall the definition of function composition

The composition \((g \circ g)(x)\) means \(g(g(x))\). So first, we need to find \(g(0)\), and then substitute that result into \(g(x)\).

Step 2: Calculate \(g(0)\)

Given \(g(x) = 2x^2 + 6\), substitute \(x = 0\):

$$ g(0) = 2(0)^2 + 6 = 0 + 6 = 6 $$

Step 3: Calculate \(g(g(0)) = g(6)\)

Given \(g(x) = 2x^2 + 6\), substitute \(x = 6\):

$$ g(6) = 2(6)^2 + 6 = 2(36) + 6 = 72 + 6 = 78 $$

Answer:

(a) \(\boldsymbol{266}\)
(b) \(\boldsymbol{398}\)
(c) \(\boldsymbol{49}\)
(d) \(\boldsymbol{78}\)