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Determine algebraically whether or not f (x) = 7x-2 and g(x)= 4x+2/7 are
inverse functions. Show how you know.
To determine whether or not f(x) = 7x - 2 and g(x) = 4x + 2/7 are inverse functions, we need to check if the composition of the two functions produces the identity function.
First, let's find the composition of f and g. To do this, we substitute g(x) into f(x) as follows:
f(g(x)) = 7(4x + 2/7) - 2
= 28x + 4/7 - 2
= 28x + 4/7 - 14/7
= 28x - 10/7
= (28x - 10)/7
Now, let's find the composition of g and f. We substitute f(x) into g(x) as follows:
g(f(x)) = 4(7x - 2) + 2/7
= 28x - 8 + 2/7
= 28x - 8 + 2/7
= 28x - 56/7 + 2/7
= 28x - 54/7
= (28x - 54)/7
If f and g are inverse functions, then f(g(x)) = x and g(f(x)) = x.
Now, let's compare f(g(x)) with x:
f(g(x)) = (28x - 10)/7
We can see that this is not equal to x.
Next, let's compare g(f(x)) with x:
g(f(x)) = (28x - 54)/7
We can see that this is not equal to x.
Since f(g(x)) and g(f(x)) are not equal to x, we can conclude that f(x) = 7x - 2 and g(x) = 4x + 2/7 are not inverse functions.