Chapter 6: Problem 3

If \(f(x)=x+6\) and \(g(x)=x^{2}-5\) find (a) \(f(g(0))\), (b) \(g(f(0))\), (c) \(g(g(2))\), (d) \(f(g(7))\).

Short Answer

Expert verified

In summary, we found the following compositions at the given specific values: a) \(f(g(0)) = 1\) b) \(g(f(0)) = 31\) c) \(g(g(2)) = -4\) d) \(f(g(7)) = 50\)

Step by step solution

Calculate g(0)

Substitute 0 into the function g(x): \(g(0) = (0)^2 - 5 = -5\)

Calculate f(g(0))

Now substitute the value of g(0) into the function f(x): \(f(g(0)) = f(-5) = (-5) + 6 = 1\) So, \(f(g(0)) = 1\). #b) g(f(0))#

Calculate f(0)

Substitute 0 into the function f(x): \(f(0) = (0) + 6 = 6\)

Calculate g(f(0))

Now substitute the value of f(0) into the function g(x): \(g(f(0)) = g(6) = (6)^2 - 5 = 36 - 5 = 31\) So, \(g(f(0)) = 31\). #c) g(g(2))#

Calculate g(2)

Substitute 2 into the function g(x): \(g(2) = (2)^2 - 5 = 4 - 5 = -1\)

Calculate g(g(2))

Now substitute the value of g(2) into the function g(x) again: \(g(g(2)) = g(-1) = (-1)^2 - 5 = 1 - 5 = -4\) So, \(g(g(2)) = -4\). #d) f(g(7))#

Calculate g(7)

Substitute 7 into the function g(x): \(g(7) = (7)^2 - 5 = 49 - 5 = 44\)

Calculate f(g(7))

Now substitute the value of g(7) into the function f(x): \(f(g(7)) = f(44) = (44) + 6 = 50\) So, \(f(g(7)) = 50\).

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If \(f(x)=x+6\) and \(g(x)=x^{2}-5\) find (a) \(f(g(0))\), (b) \(g(f(0))\), (c) \(g(g(2))\), (d) \(f(g(7))\).

Short Answer

Step by step solution

Calculate g(0)

Calculate f(g(0))

Calculate f(0)

Calculate g(f(0))

Calculate g(2)

Calculate g(g(2))

Calculate g(7)

Calculate f(g(7))

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