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Find the inverse function for f(x)=2x−6−−−−−√
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To find the inverse function of f(x) = 2x - 6^(1/2), we can follow these steps:
Step 1: Replace f(x) with y: y = 2x - 6^(1/2).
Step 2: Swap x and y: x = 2y - 6^(1/2).
Step 3: Solve for y:
x + 6^(1/2) = 2y
(x + 6^(1/2))/2 = y
Step 4: Replace y with the inverse function notation: f^(-1)(x) = (x + 6^(1/2))/2.
Therefore, the inverse function of f(x) = 2x - 6^(1/2) is f^(-1)(x) = (x + 6^(1/2))/2.
Let f (x)=1/x and g (x)=x^2+5x. A. Find (f×g)(x). B. Find the domain and range of (f×g)(x).
A. To find (f×g)(x), we need to multiply the functions f(x) and g(x). Using the given functions, f(x) = 1/x and g(x) = x^2 + 5x, we can write (f×g)(x) as f(x) * g(x):
(f×g)(x) = f(x) * g(x)
= (1/x) * (x^2 + 5x)
= (x^2 + 5x)/x
= x + 5
Therefore, (f×g)(x) is equal to x + 5.
B. To find the domain and range of (f×g)(x), we need to consider the domain and range of the individual functions f(x) and g(x).
For f(x) = 1/x, the domain is all real numbers except x = 0, because division by zero is undefined. The range is also all real numbers except y = 0, since the function cannot have a value of 0 when x ≠ 0.
For g(x) = x^2 + 5x, there are no restrictions on the domain. The range, however, depends on the shape of the parabola. Since the coefficient of the x^2 term is positive, the parabola opens upwards and the vertex is the minimum point. Hence, the range is all real numbers greater than or equal to the y-coordinate of the vertex.
To find the vertex of g(x), we can use the formula x = -b/2a for a quadratic equation in the form ax^2 + bx + c. In this case, a = 1, b = 5, and c = 0.
x = -(5)/(2*1)
x = -5/2
Substituting this value of x back into g(x), we get:
g(-5/2) = (-5/2)^2 + 5(-5/2)
= 25/4 - 25/2
= 25/4 - 50/4
= -25/4
Therefore, the range of g(x) is all real numbers greater than or equal to -25/4.
Since (f×g)(x) = x + 5, the domain is all real numbers and the range is all real numbers except y = 5.
In summary, the domain of (f×g)(x) is all real numbers and the range is all real numbers except y = 5.
To find the inverse function for f(x)=2x−6−−−−−√, we need to swap the positions of x and y and solve for y.
Step 1: Swap x and y
x = 2y - 6
Step 2: Solve for y
x + 6 = 2y
2y = x + 6
y = (x + 6) / 2
Step 3: Rewrite the equation in function notation
Now we have the inverse function: f^(-1)(x) = (x + 6) / 2