If f and g are inverse functions and if (8,3) is a point on the graph of y = g(x) then f is
a. f(x) = x^2-1 b. f(x) = 2x -1 c. f(x) = 〖2x〗^2-1. d. f(x) = 4x
If f and g are inverse functions, then f(g(x)) = x and g(f(x)) = x for all values of x in their respective domains.
Since (8,3) is a point on the graph of y = g(x), we can infer that g(8) = 3.
Using the property of inverse functions, we have f(3) = 8.
Now let's analyze each option:
a. f(x) = x^2 - 1
If we substitute 3 into this equation, we get f(3) = 3^2 - 1 = 9 - 1 = 8. This matches our earlier result, so option A could be a possible answer.
b. f(x) = 2x - 1
Substituting 3 into this equation, we get f(3) = 2(3) - 1 = 6 - 1 = 5. This does not match our earlier result, so option B is not the correct answer.
c. f(x) = 2x^2 - 1
Substituting 3 into this equation, we get f(3) = 2(3)^2 - 1 = 2(9) - 1 = 18 - 1 = 17. This does not match our earlier result, so option C is not the correct answer.
d. f(x) = 4x
Substituting 3 into this equation, we get f(3) = 4(3) = 12. This does not match our earlier result, so option D is not the correct answer.
Based on our analysis, option A (f(x) = x^2 - 1) is the most likely correct answer.
To find the inverse of a function, the coordinates need to be switched. If (8,3) is a point on the graph of y=g(x), then the inverse function would have the coordinates (3,8).
Now, let's check which answer choice matches these coordinates when plugged into f(x).
a. f(x) = x^2 - 1
Plugging in x = 3 gives: f(3) = 3^2 - 1 = 9 - 1 = 8 (not a match)
b. f(x) = 2x - 1
Plugging in x = 3 gives: f(3) = 2(3) - 1 = 6 - 1 = 5 (not a match)
c. f(x) = 2x^2 - 1
Plugging in x = 3 gives: f(3) = 2(3)^2 - 1 = 2(9) - 1 = 18 - 1 = 17 (not a match)
d. f(x) = 4x
Plugging in x = 3 gives: f(3) = 4(3) = 12 (not a match)
None of the answer choices result in f(3) being equal to 8. Therefore, none of the options above are the inverse function of g(x).
To determine which function, f(x), is the inverse of g(x), we need to use the given point (8,3) on the graph of g(x).
The inverse of a function essentially swaps the x and y values. So, if (8,3) is a point on the graph of g(x), it means that g(8) = 3.
We can find the value of x for f(x) by using the equation g(8) = 3. Since g and f are inverse functions, this equation is equivalent to f(3) = 8.
Now, let's test each answer choice by plugging in x = 3 and see which satisfies the equation f(3) = 8:
a. f(x) = x^2-1
f(3) = (3)^2 - 1 = 9 - 1 = 8 ✔️
b. f(x) = 2x - 1
f(3) = 2(3) - 1 = 6 - 1 = 5 ❌
c. f(x) = 〖2x〗^2-1
f(3) = (2(3))^2 - 1 = 6^2 - 1 = 35 ❌
d. f(x) = 4x
f(3) = 4(3) = 12 ❌
Based on the test, only choice a, f(x) = x^2-1, satisfies f(3) = 8. Therefore, the answer is a.