Question 1

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f^-1.State the domains and ranges of f and f^-1. F(x)= |x - 9| + 1

Accepted Answer

The function f(x) = |x - 9| + 1 is not one-to-one on its entire domain because it fails the horizontal line test. However, if we restrict the domain to x ≥ 9, then the function becomes one-to-one and has an inverse function.To find the inverse function, we can switch the roles of x and y and solve for y:y = |x - 9| + 1x = |y - 9| + 1|y - 9| = x - 1y - 9 = ±(x - 1)y = 9 ± (x - 1)y = 9 + x - 1 or y = 9 - x + 1y = x + 8 or y = -x + 8Since the domain of f is restricted to x ≥ 9, the inverse function f^-1(x) must be restricted to y ≥ 9. Therefore, the inverse function is f^-1(x) = x + 8.The domain of f is all real numbers x such that x ≥ 9, and the range of f is all real numbers y such that y ≥ 1. The domain of f^-1 is all real numbers x such that x ≥ 9, and the range of f^-1 is all real numbers y such that y ≥ 1.

Question 2

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f^-1.State the domains and ranges of f and f^-1. ? F(x)= -6x² + 2 ?

Accepted Answer

The correct answer is A because the inverse function is $f ^ { - 1 } ( x ) = \frac { \sqrt { - 6 ( x - 2 ) } } { 6 }$ and the domain of f and the range of f^-1 are all real numbers x such that x ? 0. The domain of f^-1 and the range of f are all real numbers x such that x ? 2.

Question 3

Determine whether the function has an inverse function.If it does,find the inverse function. ? F(x)= (x + 4)²,x ? -4 ?

Accepted Answer

The function $F(x) = (x + 4)^2, x \geq -4$ is a one-to-one function on the domain $x \geq -4$, so it has an inverse. To find the inverse, replace $F(x)$ with $y$, then solve for $x$: $y = (x + 4)^2$. Taking the square root of both sides gives $x = \sqrt{y} - 4$, so the inverse function is $f^{-1}(x) = \sqrt{x} - 4$.

Question 4

Your wage is $11.00 per hour plus $0.50 for each unit produced per hour.So,your hourly wage in terms of the number of units produced x is y = 11 + 0.50x.Find the inverse function.What does each variable represent in the inverse function? ?

Accepted Answer

The inverse function is found by solving the original equation for x, which represents the number of units produced, and y, which now represents the hourly wage.

Question 5

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f^-1.State the domains and ranges of f and f^-1. ? F(x)= (x - 5)² ?

Accepted Answer

Given $f(x) = (x - 5)^2$, the function is one-to-one if we restrict the domain to $x \geq 5$, ensuring it's increasing on this interval. The inverse function is found by solving $y = (x - 5)^2$ for $x$, giving $x = \sqrt{y} + 5$, thus $f^{-1}(x) = \sqrt{x} + 5$. The domain of $f$ and the range of $f^{-1}$ are $x \geq 5$, and the domain of $f^{-1}$ and the range of $f$ are $x \geq 0$, matching option C.

Question 6

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 5$ and g(x)= x³ to find (f^-1 _º f^-1)(-5). ?

Accepted Answer

First, find the inverse of $f(x)$: $f^{-1}(x) = 8(x + 5)$. Then, calculate $f^{-1}(f^{-1}(-5))$: $f^{-1}(-5) = 8(-5 + 5) = 0$, and $f^{-1}(0) = 8(0 + 5) = 40$.

Question 7

Use the functions given by f(x)= x + 6 and g(x)= 7x - 3 to find g^-1 _º f^-1. ?

Accepted Answer

The answer of Use the functions given by f(x)= x...

Question 8

Determine whether the function has an inverse function.If it does,find the inverse function.? $h ( x ) = - \frac { 4 } { x ^ { 2 } }$ ?

Accepted Answer

The function $h(x) = -\frac{4}{x^2}$ does not pass the horizontal line test because for any given y-value (except 0), there are two x-values that produce that y-value (one positive and one negative), indicating that the function is not one-to-one and therefore does not have an inverse.

Question 9

Use the functions given by $f ( x ) = \frac { 1 } { 125 } x - 1$ and g(x)= x³ to find (f _º g)^-1. ?

Accepted Answer

The answer of Use the functions given by \(f (...

Question 10

Use the functions given by $f ( x ) = \frac { 1 } { 64 } x - 4$ and g(x)= x³ to find g^-1 _º f^-1. ?

Accepted Answer

Given $f(x) = \frac{1}{64}x - 4$ and $g(x) = x^{13}$, to find $g^{-1}(f^{-1}(x))$, we first find the inverse functions of $f$ and $g$. The inverse of $f(x)$, $f^{-1}(x)$, is found by solving $y = \frac{1}{64}x - 4$ for $x$, which gives $x = 64(y + 4)$. The inverse of $g(x) = x^{13}$, $g^{-1}(x)$, is $x^{1/13}$. Substituting $f^{-1}(x)$ into $g^{-1}(x)$ gives $g^{-1}(f^{-1}(x)) = (64(x + 4))^{1/13} = 4\sqrt[3]{x + 4}$, which matches choice E.

Question 11

Determine whether the function has an inverse function.If it does,find the inverse function.? $f ( x ) = \sqrt { 7 x + 8 }$ ?

Accepted Answer

The answer of Determine whether the function has an inverse...

Question 12

Determine whether the function has an inverse function.If it does,find the inverse function. ?
F(x)= -2
?

Accepted Answer

The function $F(x) = -2$ is a constant function, meaning it maps every input to the same output, -2. For a function to have an inverse, each input must map to a unique output. Since this is not the case for a constant function, it does not have an inverse.

Question 13

Determine whether the function has an inverse function.If it does,find the inverse function.? $g ( x ) = \frac { x } { 5 }$ ?

Accepted Answer

To find the inverse function, swap $x$ and $y$ and solve for $y$. Starting with $y = \frac{x}{5}$, swapping gives $x = \frac{y}{5}$, and solving for $y$ gives $y = 5x$, which is the inverse function, $g^{-1}(x) = 5x$.

Question 14

Use the functions given by f(x)= x + 2 and g(x)= 2x - 5 to find (f _º g)^-1. ?

Accepted Answer

The composition of functions $f(g(x))$ is calculated by substituting $g(x)$ into $f(x)$, resulting in $f(g(x)) = (2x - 5) + 2 = 2x - 3$. The inverse of this function, $(f \circ g)^{-1}(x)$, involves swapping $x$ and $y$ and solving for $y$, leading to $x = 2y - 3$, which simplifies to $y = \frac{x + 3}{2}$.

Question 15

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f^-1.State the domains and ranges of f and f^-1. F(x)= |x + 5|

Accepted Answer

To ensure the function $f(x) = |x + 5|$ is one-to-one, we restrict its domain to $x \geq -5$, where it behaves like $f(x) = x + 5$. The inverse function is $f^{-1}(x) = x - 5$, with the domain of $f$ and range of $f^{-1}$ being $x \geq -5$, and the domain of $f^{-1}$ and range of $f$ being $x \geq 0$.

Question 16

?Use the graph of the function f to create a table of values for the given points.Then create a second table that can be used to find f^-1. ?? ?

Accepted Answer

The answer of ?Use the graph of the function f...

Question 17

The function given by y = 0.03x² + 245.50,0 < x < 100 approximates the exhaust temperature y in degrees Fahrenheit,x where is the percent load for a diesel engine.Find the inverse function. ?

Accepted Answer

To find the inverse, solve for x: $ y = 0.03x^2 + 245.50 $. Subtract 245.50 from both sides, then divide by 0.03, and finally take the square root.

Question 18

Determine whether the function has an inverse function.If it does,find the inverse function.? $f ( x ) = \left\{ \begin{array} { l } 
x + 2 , x < 0 \
2 - x , x \geq 0
\end{array} $ ?

Accepted Answer

The function $f(x)$ is not one-to-one because it is not strictly increasing or decreasing over its entire domain. Therefore, it does not have an inverse function.

Question 19

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 5$ and g(x)= x³ to find (g^-1 _º f^-1)(-5). ?

Accepted Answer

The answer of Use the functions given by \(f (...

Question 20

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 1$ and g(x)= x³ to find (f^-1 _º g^-1)(1). ?

Accepted Answer

The answer of Use the functions given by \(f (...

Quiz 9: Inverse Functions

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f^-1.State the domains and ranges of f and f^-1. F(x) = |x - 9| + 1

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f^-1.State the domains and ranges of f and f^-1. ? F(x) = -6x² + 2 ?

Determine whether the function has an inverse function.If it does,find the inverse function. ? F(x) = (x + 4) ²,x ? -4 ?

Your wage is $11.00 per hour plus $0.50 for each unit produced per hour.So,your hourly wage in terms of the number of units produced x is y = 11 + 0.50x.Find the inverse function.What does each variable represent in the inverse function? ?

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f^-1.State the domains and ranges of f and f^-1. ? F(x) = (x - 5) ² ?

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 5$ and g(x) = x³ to find (f^-1 _º f^-1) (-5) . ?

Use the functions given by f(x) = x + 6 and g(x) = 7x - 3 to find g^-1 _º f^-1. ?

Determine whether the function has an inverse function.If it does,find the inverse function.? $h ( x ) = - \frac { 4 } { x ^ { 2 } }$ ?

Use the functions given by $f ( x ) = \frac { 1 } { 125 } x - 1$ and g(x) = x³ to find (f _º g) ^-1. ?

Use the functions given by $f ( x ) = \frac { 1 } { 64 } x - 4$ and g(x) = x³ to find g^-1 _º f^-1. ?

Determine whether the function has an inverse function.If it does,find the inverse function.? $f ( x ) = \sqrt { 7 x + 8 }$ ?

Determine whether the function has an inverse function.If it does,find the inverse function. ? F(x) = -2 ?

Determine whether the function has an inverse function.If it does,find the inverse function.? $g ( x ) = \frac { x } { 5 }$ ?

Use the functions given by f(x) = x + 2 and g(x) = 2x - 5 to find (f _º g) ^-1. ?

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f^-1.State the domains and ranges of f and f^-1. F(x) = |x + 5|

The function given by y = 0.03x² + 245.50,0 < x < 100 approximates the exhaust temperature y in degrees Fahrenheit,x where is the percent load for a diesel engine.Find the inverse function. ?

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 5$ and g(x) = x³ to find (g^-1 _º f^-1) (-5) . ?

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 1$ and g(x) = x³ to find (f^-1 _º g^-1) (1) . ?

Quiz 9: Inverse Functions

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f-1.State the domains and ranges of f and f-1. ​ F(x) = |x - 9| + 1 ​

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f-1.State the domains and ranges of f and f-1. ? F(x) = -6x2 + 2 ?

Determine whether the function has an inverse function.If it does,find the inverse function. ? F(x) = (x + 4) 2,x ? -4 ?

Your wage is $11.00 per hour plus $0.50 for each unit produced per hour.So,your hourly wage in terms of the number of units produced x is y = 11 + 0.50x.Find the inverse function.What does each variable represent in the inverse function? ?

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f-1.State the domains and ranges of f and f-1. ? F(x) = (x - 5) 2 ?

Use the functions given by f(x) =18x−5f ( x ) = \frac { 1 } { 8 } x - 5f(x) =81​x−5 and g(x) = x3 to find (f-1 º f-1) (-5) . ?

Use the functions given by f(x) = x + 6 and g(x) = 7x - 3 to find g-1 º f-1. ?

Determine whether the function has an inverse function.If it does,find the inverse function.? h(x) =−4x2h ( x ) = - \frac { 4 } { x ^ { 2 } }h(x) =−x24​ ?

Use the functions given by f(x) =1125x−1f ( x ) = \frac { 1 } { 125 } x - 1f(x) =1251​x−1 and g(x) = x3 to find (f º g) -1. ?

Use the functions given by f(x) =164x−4f ( x ) = \frac { 1 } { 64 } x - 4f(x) =641​x−4 and g(x) = x3 to find g-1 º f-1. ?

Determine whether the function has an inverse function.If it does,find the inverse function.? f(x) =7x+8f ( x ) = \sqrt { 7 x + 8 }f(x) =7x+8​ ?

Determine whether the function has an inverse function.If it does,find the inverse function. ? F(x) = -2 ?

Determine whether the function has an inverse function.If it does,find the inverse function.? g(x) =x5g ( x ) = \frac { x } { 5 }g(x) =5x​ ?

Use the functions given by f(x) = x + 2 and g(x) = 2x - 5 to find (f º g) -1. ?

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f-1.State the domains and ranges of f and f-1. ​ F(x) = |x + 5| ​

?Use the graph of the function f to create a table of values for the given points.Then create a second table that can be used to find f-1. ?? ?

The function given by y = 0.03x2 + 245.50,0 < x < 100 approximates the exhaust temperature y in degrees Fahrenheit,x where is the percent load for a diesel engine.Find the inverse function. ?

Use the functions given by f(x) =18x−5f ( x ) = \frac { 1 } { 8 } x - 5f(x) =81​x−5 and g(x) = x3 to find (g-1 º f-1) (-5) . ?

Use the functions given by f(x) =18x−1f ( x ) = \frac { 1 } { 8 } x - 1f(x) =81​x−1 and g(x) = x3 to find (f-1 º g-1) (1) . ?

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f^-1.State the domains and ranges of f and f^-1. F(x) = |x - 9| + 1

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f^-1.State the domains and ranges of f and f^-1. ? F(x) = -6x² + 2 ?

Determine whether the function has an inverse function.If it does,find the inverse function. ? F(x) = (x + 4) ²,x ? -4 ?

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f^-1.State the domains and ranges of f and f^-1. ? F(x) = (x - 5) ² ?

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 5$ and g(x) = x³ to find (f^-1 _º f^-1) (-5) . ?

Use the functions given by f(x) = x + 6 and g(x) = 7x - 3 to find g^-1 _º f^-1. ?

Determine whether the function has an inverse function.If it does,find the inverse function.? $h ( x ) = - \frac { 4 } { x ^ { 2 } }$ ?

Use the functions given by $f ( x ) = \frac { 1 } { 125 } x - 1$ and g(x) = x³ to find (f _º g) ^-1. ?

Use the functions given by $f ( x ) = \frac { 1 } { 64 } x - 4$ and g(x) = x³ to find g^-1 _º f^-1. ?

Determine whether the function has an inverse function.If it does,find the inverse function.? $f ( x ) = \sqrt { 7 x + 8 }$ ?

Determine whether the function has an inverse function.If it does,find the inverse function.? $g ( x ) = \frac { x } { 5 }$ ?

Use the functions given by f(x) = x + 2 and g(x) = 2x - 5 to find (f _º g) ^-1. ?

Restrict the domain of the function f so that the function is one-to-one and has an inverse function.Then find the inverse function f^-1.State the domains and ranges of f and f^-1. F(x) = |x + 5|

The function given by y = 0.03x² + 245.50,0 < x < 100 approximates the exhaust temperature y in degrees Fahrenheit,x where is the percent load for a diesel engine.Find the inverse function. ?

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 5$ and g(x) = x³ to find (g^-1 _º f^-1) (-5) . ?

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 1$ and g(x) = x³ to find (f^-1 _º g^-1) (1) . ?