Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Sure! Let's find the inverse of each function step by step.
### Finding the inverse of [tex]\( f(x) = -2x + 11 \)[/tex]
1. Start with the function:
[tex]\[ f(x) = -2x + 11 \][/tex]
2. To find the inverse, we first replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = -2x + 11 \][/tex]
3. Next, interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = -2y + 11 \][/tex]
4. Solve this equation for [tex]\( y \)[/tex]:
[tex]\[ x = -2y + 11 \][/tex]
[tex]\[ x - 11 = -2y \][/tex]
[tex]\[ -2y = x - 11 \][/tex]
[tex]\[ y = \frac{11 - x}{2} \][/tex]
5. Replace [tex]\( y \)[/tex] with [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = \frac{11 - x}{2} \][/tex]
So, the inverse function of [tex]\( f(x) = -2x + 11 \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{11 - x}{2} \][/tex]
### Finding the inverse of [tex]\( g(x) = \frac{4}{x + 7} \)[/tex]
1. Start with the function:
[tex]\[ g(x) = \frac{4}{x + 7} \][/tex]
2. To find the inverse, replace [tex]\( g(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \frac{4}{x + 7} \][/tex]
3. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \frac{4}{y + 7} \][/tex]
4. Solve this equation for [tex]\( y \)[/tex]:
[tex]\[ x(y + 7) = 4 \][/tex]
[tex]\[ xy + 7x = 4 \][/tex]
[tex]\[ xy = 4 - 7x \][/tex]
[tex]\[ y = \frac{4 - 7x}{x} \][/tex]
5. Replace [tex]\( y \)[/tex] with [tex]\( g^{-1}(x) \)[/tex]:
[tex]\[ g^{-1}(x) = \frac{4 - 7x}{x} \][/tex]
So, the inverse function of [tex]\( g(x) = \frac{4}{x + 7} \)[/tex] is:
[tex]\[ g^{-1}(x) = \frac{4 - 7x}{x} \][/tex]
In summary:
1. The inverse of [tex]\( f(x) = -2x + 11 \)[/tex] is [tex]\( f^{-1}(x) = \frac{11 - x}{2} \)[/tex].
2. The inverse of [tex]\( g(x) = \frac{4}{x + 7} \)[/tex] is [tex]\( g^{-1}(x) = \frac{4 - 7x}{x} \)[/tex].
### Finding the inverse of [tex]\( f(x) = -2x + 11 \)[/tex]
1. Start with the function:
[tex]\[ f(x) = -2x + 11 \][/tex]
2. To find the inverse, we first replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = -2x + 11 \][/tex]
3. Next, interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = -2y + 11 \][/tex]
4. Solve this equation for [tex]\( y \)[/tex]:
[tex]\[ x = -2y + 11 \][/tex]
[tex]\[ x - 11 = -2y \][/tex]
[tex]\[ -2y = x - 11 \][/tex]
[tex]\[ y = \frac{11 - x}{2} \][/tex]
5. Replace [tex]\( y \)[/tex] with [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = \frac{11 - x}{2} \][/tex]
So, the inverse function of [tex]\( f(x) = -2x + 11 \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{11 - x}{2} \][/tex]
### Finding the inverse of [tex]\( g(x) = \frac{4}{x + 7} \)[/tex]
1. Start with the function:
[tex]\[ g(x) = \frac{4}{x + 7} \][/tex]
2. To find the inverse, replace [tex]\( g(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \frac{4}{x + 7} \][/tex]
3. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \frac{4}{y + 7} \][/tex]
4. Solve this equation for [tex]\( y \)[/tex]:
[tex]\[ x(y + 7) = 4 \][/tex]
[tex]\[ xy + 7x = 4 \][/tex]
[tex]\[ xy = 4 - 7x \][/tex]
[tex]\[ y = \frac{4 - 7x}{x} \][/tex]
5. Replace [tex]\( y \)[/tex] with [tex]\( g^{-1}(x) \)[/tex]:
[tex]\[ g^{-1}(x) = \frac{4 - 7x}{x} \][/tex]
So, the inverse function of [tex]\( g(x) = \frac{4}{x + 7} \)[/tex] is:
[tex]\[ g^{-1}(x) = \frac{4 - 7x}{x} \][/tex]
In summary:
1. The inverse of [tex]\( f(x) = -2x + 11 \)[/tex] is [tex]\( f^{-1}(x) = \frac{11 - x}{2} \)[/tex].
2. The inverse of [tex]\( g(x) = \frac{4}{x + 7} \)[/tex] is [tex]\( g^{-1}(x) = \frac{4 - 7x}{x} \)[/tex].
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.