: Determine the equation of inverse f(x) = 2x^2+24-3. What is the domain and range of the inverse function.
Probably contains a typo, was -24 supposed to be -24x ?
Poorly worded, did you mean:
Determine the equation of the inverse of f(x) = 2x^2 + 24x - 3 ?
If so, the inverse of f(x) is not a function.
Waiting for a cleanup.
Yes it's quadratic 24x.R
To find the inverse of a function f(x), we need to switch the roles of x and y and solve for y.
Step 1: Replace f(x) with y.
f(x) = 2x^2 + 24 - 3
y = 2x^2 + 21
Step 2: Swap x and y.
x = 2y^2 + 21
Step 3: Solve for y.
x - 21 = 2y^2
(x - 21) / 2 = y^2
±√((x - 21) / 2) = y
The equation of the inverse function is f^(-1)(x) = ±√((x - 21) / 2).
The domain of the inverse function depends on the domain of the original function. Since f(x) is a quadratic function, it is defined for all real numbers. Hence, the domain of f^(-1)(x) is also all real numbers.
The range of the inverse function can be determined by looking at the range of the original function. Since f(x) = 2x^2 + 24 - 3 is a quadratic function with a positive leading coefficient, its minimum point is at the vertex. The vertex occurs at (h, k), where h = -b / (2a) and k = f(h).
In this case, a = 2, b = 0, and c = 21. So, h = -0 / (2 * 2) = 0, and k = f(0) = 2(0)^2 + 24 - 3 = 21.
Since the minimum point of f(x) is at (0, 21), the range of f(x) is [21, ∞). Therefore, the range of f^(-1)(x) is also [21, ∞).
To find the equation of the inverse function, we need to perform the following steps:
Step 1: Start with the original function f(x) = 2x^2 + 24 - 3.
Step 2: Replace f(x) with y to represent the function in terms of y: y = 2x^2 + 24 - 3.
Step 3: Swap the variables x and y: x = 2y^2 + 24 - 3.
Step 4: Rearrange the equation to solve for y: 2y^2 = x - 21.
Step 5: Divide both sides of the equation by 2: y^2 = (x - 21) / 2.
Step 6: Take the square root of both sides to isolate y: y = ±√((x - 21) / 2).
Therefore, the equation of the inverse function is f^(-1)(x) = ±√((x - 21) / 2).
To determine the domain and range of the inverse function, we need to consider the restrictions on the original function f(x) = 2x^2 + 24 - 3:
Domain: The original function f(x) = 2x^2 + 24 - 3 is a quadratic function, which is defined for all real numbers. Therefore, the domain of the inverse function f^(-1)(x) will also be all real numbers.
Range: The range of the original function f(x) = 2x^2 + 24 - 3 can be determined by analyzing the vertex of the parabola. Since the coefficient of the x^2 term is positive, the parabola opens upwards, and the vertex represents the minimum point. The x-coordinate of the vertex is given by -b/2a, where a and b are the coefficients of the quadratic equation. In this case, a = 2 and b = 0, so the x-coordinate of the vertex is x = 0 / (2 * 2) = 0. Substituting this x-value back into the original function, we get f(0) = 2(0)^2 + 24 - 3 = 21. Therefore, the range of the original function f(x) is [21, ∞).
Since the domain and range of the original function are switched when finding the inverse function, the domain of the inverse function will be [21, ∞), and the range will be all real numbers.