If f(x)=x^2 –4, ... x > or equal to 0
and g(x)=sqrt(x+4) ... x > or equal to –4 ,
(a) f(g(x))=
(b) g(f(x))=
(c) Thus g(x) is called an INVERSE function of f(x)
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If f is one-to-one and f(–4)=14, then
f^–1 (14)=
and (f(–4))^–1= .
If g is one-to-one and g(4)=4, then
g^–1 (4)=
and (g(4))^–1= .
If h is one-to-one and h(1)=14, then
h^–1 (14)=
and (h(1)^)–1=
To find the compositions of functions and inverse functions, we can follow these steps:
(a) To find f(g(x)), we substitute g(x) into f(x):
f(g(x)) = (g(x))^2 - 4
First, substitute g(x) into f(x):
g(x) = sqrt(x + 4)
Now, substitute g(x) into f(x):
f(g(x)) = (sqrt(x + 4))^2 - 4
Simplify the expression:
f(g(x)) = (x + 4) - 4
f(g(x)) = x
Therefore, f(g(x)) = x.
(b) To find g(f(x)), we substitute f(x) into g(x):
g(f(x)) = sqrt(f(x) + 4)
First, substitute f(x) into g(x):
f(x) = x^2 - 4
Now, substitute f(x) into g(x):
g(f(x)) = sqrt((x^2 - 4) + 4)
Simplify the expression:
g(f(x)) = sqrt(x^2)
g(f(x)) = |x|
Therefore, g(f(x)) = |x|.
(c) To show that g(x) is the inverse function of f(x), we can verify that the composition of g(f(x)) is the identity function.
From part (b), we found that g(f(x)) = |x|. The absolute value function returns the original input, so g(f(x)) simplifies to x. This shows that g(x) "undoes" the operation performed by f(x), making g(x) the inverse function of f(x).
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To find the inverse of a one-to-one function, we can follow these steps:
For the function f:
1. Start with the equation f(x) = some value.
2. Replace f(x) with y: y = f(x).
3. Swap x and y: x = f^(-1)(y).
4. Solve for y: f^(-1)(y) = x.
5. Replace y with the given value: f^(-1)(14) = x.
Specifically, for f(–4) = 14:
1. Start with -4 = 14.
2. Replace -4 with y: y = 14.
3. Swap x and y: 14 = f^(-1)(x).
4. Solve for y: f^(-1)(14) = -4.
Therefore, f^(-1)(14) = -4.
And (f(–4))^(-1) refers to the inverse of the value obtained from f(-4).
For f(–4) = 14:
1. Start with -4 = 14.
2. Raise both sides to the power of -1: (-4)^(-1) = 14^(-1).
3. Evaluate the expressions: -1/4 = 1/14.
Therefore, (f(–4))^(-1) = -1/4.
Similarly, you can follow the same steps to find the inverse of function g and h.