1. To find the inverse function for f(x)= sqrt(2x-6), we follow the steps below:
- Let y = sqrt(2x-6)
- Swap x and y: x = sqrt(2y-6)
- Solve for y: x^2 = 2y-6
- Add 6 to both sides: x^2 + 6 = 2y
- Divide by 2: (x^2 + 6)/2 = y
- Therefore, the inverse function is f^(-1)(x) = (x^2 + 6)/2
2.a. To show that the function f(x) = (x-2)^3 + 8 is one-to-one algebraically, we need to show that for any two different values of x, the values of f(x) are also different.
- Assume x1 and x2 are two different values of x.
- Then, f(x1) = (x1-2)^3 + 8 and f(x2) = (x2-2)^3 + 8.
- If f(x1) = f(x2), then (x1-2)^3 + 8 = (x2-2)^3 + 8.
- Simplifying the equation gives us (x1-2)^3 = (x2-2)^3.
- Taking the cube root of both sides gives us x1 - 2 = x2 - 2.
- Simplifying further gives us x1 = x2, which contradicts our assumption that x1 and x2 are different.
- Therefore, f(x) = (x-2)^3 + 8 is one-to-one.
b. To find the inverse of f(x) = (x-2)^3 + 8, we follow the steps below:
- Let y = (x-2)^3 + 8.
- Swap x and y: x = (y-2)^3 + 8.
- Solve for y: x-8 = (y-2)^3.
- Take the cube root of both sides: (x-8)^(1/3) = y-2.
- Add 2 to both sides: (x-8)^(1/3) + 2 = y.
- Therefore, the inverse function is f^(-1)(x) = (x-8)^(1/3) + 2.
3.a. To find (f · g)(x), we first need to plug g(x) into f(x):
- (f · g)(x) = f(g(x)) = f(x^2 + 5x).
- Plugging the function g(x) = x^2 + 5x into f(x), we get:
- f(g(x)) = f(x^2 + 5x) = 1/(x^2 + 5x).
b. To find the domain and range of (f · g)(x) = 1/(x^2 + 5x), we consider the restrictions:
- The denominator, x^2 + 5x, cannot be equal to 0 for the function to be defined.
- Solving x^2 + 5x = 0 gives us x(x+5) = 0, which gives us two possible values for x: x = 0 and x = -5.
- Therefore, the domain of (f · g)(x) is all real numbers except x = 0 and x = -5.
- The range of (f · g)(x) is all real numbers except y = 0, since the denominator cannot be 0.
1. Find the inverse function for f(x)=sqr 2x-6
2. Let f(x)=(x-2)^3+8
a. Show that this function is one-to-one algebraically.
b. Find the inverse of f(x).
3. 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)
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