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.
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?
7 answers
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
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; \(f^{-1}(x) = -2 + \sqrt{x+1}\)
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.
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
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
d) g^(-1)(x) = x^2 + 7, x ≥ 0