For the function lower f left-parenthesis x right-parenthesis equals negative 4 Start Root x End Root minus 1, find the inverse function.
To find the inverse function of f(x) = -4√x - 1, we will follow these steps:
Step 1: Replace f(x) with y.
y = -4√x - 1
Step 2: Swap x and y.
x = -4√y - 1
Step 3: Solve the equation for y.
x + 1 = -4√y
Divide both sides by -4.
-1/4(x + 1) = √y
Square both sides.
(1/16)(x^2 + 2x + 1) = y
Step 4: Replace y with f^(-1)(x)
f^(-1)(x) = (1/16)(x^2 + 2x + 1)
Therefore, the inverse function of f(x) = -4√x - 1 is f^(-1)(x) = (1/16)(x^2 + 2x + 1).
choices
a) lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction, x less-than-or-equal-to negative 1Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction, x less-than-or-equal-to negative 1
b) lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction, x greater-than-or-equal-to negative 1
Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction, x greater-than-or-equal-to negative 1
c) lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction
Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction
d) lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals Start Fraction left-parenthesis x plus 1 right-parenthesis squared over 16 End Fraction, x less-than-or-equal-to negative 1
The correct answer is:
c) f^(-1)(x) = (1/16)(x + 1)^2
The function lower f left-parenthesis x right-parenthesis equals left-parenthesis x minus 1 right-parenthesis squared minus 4 is not one-to-one. If you restrict the domain for f left-parenthesis x right-parenthesis to x less-than-or-equal-to 1, what is its inverse function and the domain for the inverse?
If we restrict the domain of f(x) to x ≤ 1, we can see that the function becomes one-to-one and we can find its inverse using the following steps:
Step 1: Replace f(x) with y.
y = (x - 1)^2 - 4
Step 2: Swap x and y.
x = (y - 1)^2 - 4
Step 3: Solve the equation for y.
x + 4 = (y - 1)^2
Take the square root of both sides (ignoring the negative root since y can't be negative in this case).
√(x + 4) = y - 1
Add 1 to both sides.
y = √(x + 4) + 1
Step 4: Replace y with f^(-1)(x).
f^(-1)(x) = √(x + 4) + 1
The inverse function is f^(-1)(x) = √(x + 4) + 1. The domain for the inverse function is x ≥ -4, since we can only take the square root of a non-negative value.
choices
a) lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 minus Start Root x plus 4 End Root, x greater-than-or-equal-to negative 4Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 minus Start Root x plus 4 End Root, x greater-than-or-equal-to negative 4
b) lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 plus Start Root x plus 4 End Root, x greater-than-or-equal-to negative 4
Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 plus Start Root x plus 4 End Root, x greater-than-or-equal-to negative 4
c) lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 minus Start Root x minus 4 End Root, x greater-than-or-equal-to 4
Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 minus Start Root x minus 4 End Root, x greater-than-or-equal-to 4
d) lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 plus Start Root x minus 4 End Root, x greater-than-or-equal-to 4
The correct answer is:
a) f^(-1)(x) = 1 - √(x + 4), x ≥ -4