For each pair of functions f and g below, find f(g(x)) and g(f(x)).
1 answer:
Answer:
a) f(g(x)) = =
= x
g(f(x)) = =
= x
f and g are inverses
b) f(g(x)) = x + 3 + 3 = x + 6
g(f(x)) = x + 3 + 3 = x + 6
f and g are not inverses
Step-by-step explanation:
a)
f(g(x)) = =
= x
g(f(x)) = =
= x
Since f(g(x)) = g(f(x)) = x, then f and g are inverses
b)
f(g(x)) = x + 3 + 3 = x + 6
g(f(x)) = x + 3 + 3 = x + 6
Since f(g(x)) = g(f(x)) ≠ x, then f and g are not inverses
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Explanation
We must the tangent line at x = 3 of the function:
The tangent line is given by:
Where:
• m is the slope of the tangent line of f(x) at x = h,
,• k = f(h) is the value of the function at x = h.
In this case, we have h = 3.
1) First, we compute the derivative of f(x):
2) By evaluating the result of f'(x) at x = h = 3, we get:
3) The value of k is:
4) Replacing the values of m, h and k in the general equation of the tangent line, we get:
Plotting the function f(x) and the tangent line we verify that our result is correct:
AnswerThe equation of the tangent line to f(x) and x = 3 is: