The function lower f left-parenthesis x right-parenthesis equals left-parenthesis x plus 2 right-parenthesis squared minus 1 is not one-to-one. Restrict the function’s domain and find its inverse. What is the function’s restricted domain and what is its inverse function?
To find the restricted domain of the function, we need to consider the values of x that make the function not one-to-one.
The function f(x) = (x+2)^2 - 1 is a quadratic function, which means it has a U-shape graph. Since the vertex of the U shape is at the point (-2, -1), any x-value less than -2 or greater than -2 will have a corresponding y-value of -1. This means that any two different x-values from these two regions will result in the same y-value, and hence the function is not one-to-one in these regions.
Therefore, we can restrict the domain of the function to -∞ < x < -2 or -2 < x < ∞.
To find the inverse of the function, we can follow these steps:
1. Replace f(x) with y: y = (x+2)^2 - 1.
2. Swap x and y: x = (y+2)^2 - 1.
3. Solve for y: x = y^2 + 4y + 4 - 1.
Simplify: x = y^2 + 4y + 3.
4. Rearrange the equation: y^2 + 4y + 3 - x = 0.
5. Solve for y using the quadratic formula:
y = (-4 ± sqrt(4^2 - 4(1)(3-x))) / (2*1).
Simplify: y = (-4 ± sqrt(16 - 12 + 4x)) / 2.
y = (-4 ± sqrt(4x + 4)) / 2.
6. Simplify: y = (-2 ± sqrt(x + 1)).
7. So, the inverse function of f(x) is f^(-1)(x) = (-2 ± sqrt(x + 1)).
Note that since we restricted the domain of f(x) to certain intervals, the inverse function will also be restricted to those intervals.
choices
a) restricted domain: x greater-than-or-equal-to negative 2; lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals negative 2 minus Start Root x plus 1 End Root
b) restricted domain: x greater-than-or-equal-to negative 2; lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals negative 2 plus Start Root x plus 1 End Root
c) restricted domain: x greater-than-or-equal-to 0; lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals negative 2 plus Start Root x plus 1 End Root
d) restricted domain: x greater-than-or-equal-to negative 2; lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 2 plus Start Root x plus 1 End Root
The correct answer is:
b) restricted domain: x ≥ -2; \(f^{-1}(x) = -2 + \sqrt{x+1}\)
Determine the inverse of the function lower g left-parenthesis x right-parenthesis equals Start Root x minus 7 End Root
To find the inverse of the function g(x) = √(x-7), we can follow these steps:
1. Replace g(x) with y: y = √(x-7).
2. Swap x and y: x = √(y-7).
3. Square both sides of the equation to eliminate the square root: x^2 = y - 7.
4. Solve for y: y = x^2 + 7.
So, the inverse function of g(x) is g^(-1)(x) = x^2 + 7.
choices
a) lower g superscript negative 1 baseline left-parenthesis x right-parenthesis equals x squared plus 14 x plus 49, x less-than-or-equal-to 0Image with alt text: lower g superscript negative 1 baseline left-parenthesis x right-parenthesis equals x squared plus 14 x plus 49, x less-than-or-equal-to 0
b) lower g superscript negative 1 baseline left-parenthesis x right-parenthesis equals x squared plus 14 x plus 49, x greater-than-or-equal-to 0
Image with alt text: lower g superscript negative 1 baseline left-parenthesis x right-parenthesis equals x squared plus 14 x plus 49, x greater-than-or-equal-to 0
c) lower g superscript negative 1 baseline left-parenthesis x right-parenthesis equals x squared plus 7, x less-than-or-equal-to 0
Image with alt text: lower g superscript negative 1 baseline left-parenthesis x right-parenthesis equals x squared plus 7, x less-than-or-equal-to 0
d) lower g superscript negative 1 baseline left-parenthesis x right-parenthesis equals x squared plus 7, x greater-than-or-equal-to 0
The correct answer is:
d) g^(-1)(x) = x^2 + 7, x ≥ 0