a. To find the inverse of the function f(x) = x^2n + x^n + 1, we can rearrange the equation and solve for x.
Step 1: Start with the equation f(x) = x^2n + x^n + 1.
Step 2: Replace f(x) with y, so the equation becomes y = x^2n + x^n + 1.
Step 3: Swap x and y in the equation, so we have x = y^2n + y^n + 1.
Step 4: Now, solve for y by rearranging the equation:
x - 1 = y^2n + y^n.
Step 5: We can see that this equation is not easily solvable for y. Therefore, the inverse of f(x) = x^2n + x^n + 1 does not exist for the largest possible domain.
Domain of f(x): All real numbers.
Range of f(x): All real numbers greater than or equal to 1.
Domain of f^(-1)(x): Not defined (since the inverse does not exist).
Range of f^(-1)(x): Not defined.
b. To find the inverse of the function g(x) = (x^2 - 4)^(1/2), we can follow a similar process.
Step 1: Start with the equation g(x) = (x^2 - 4)^(1/2).
Step 2: Replace g(x) with y, so the equation becomes y = (x^2 - 4)^(1/2).
Step 3: Swap x and y in the equation, so we have x = (y^2 - 4)^(1/2).
Step 4: Now, solve for y by rearranging the equation:
y = (x^2 - 4)^(1/2).
Step 5: Here, the range of the function g(x) is non-negative numbers (including zero), since taking the square root results in positive values. Therefore, the domain of the inverse function f^(-1)(x) should be non-negative numbers.
Domain of g(x): All real numbers.
Range of g(x): Non-negative numbers (including zero).
Domain of g^(-1)(x): Non-negative numbers (including zero).
Range of g^(-1)(x): All real numbers.
c. To find the inverse of the function h(x) = x/(x^3), we can proceed as follows:
Step 1: Start with the equation h(x) = x/(x^3).
Step 2: Replace h(x) with y, so the equation becomes y = x/(x^3).
Step 3: Swap x and y in the equation, so we have x = y/(y^3).
Step 4: Now, solve for y by rearranging the equation:
y = 1/x^2.
Step 5: Here, the domain of the function h(x) does not include x = 0, as it would cause division by zero. Therefore, the domain of the inverse function h^(-1)(x) should exclude x = 0.
Domain of h(x): All real numbers except x = 0.
Range of h(x): All real numbers.
Domain of h^(-1)(x): All real numbers except x = 0.
Range of h^(-1)(x): All real numbers.
d. To find the inverse of the function S(x) = 1/2 (e^x - e^(-x)), we can follow a similar process.
Step 1: Start with the equation S(x) = 1/2 (e^x - e^(-x)).
Step 2: Replace S(x) with y, so the equation becomes y = 1/2 (e^x - e^(-x)).
Step 3: Swap x and y in the equation, so we have x = 1/2 (e^y - e^(-y)).
Step 4: Now, solve for y by rearranging the equation:
2x = e^y - e^(-y).
Step 5: Unfortunately, there is no simple algebraic way to solve this equation for y. Therefore, the inverse of S(x) = 1/2 (e^x - e^(-x)) does not have an easily expressible form.
Domain of S(x): All real numbers.
Range of S(x): All real numbers.
Domain of S^(-1)(x): Not defined (since the inverse does not exist).
Range of S^(-1)(x): Not defined.