Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Get quick and reliable solutions to your questions from a community of experienced experts on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Sure, here's a detailed step-by-step solution to the problem:
### (a) Find
#### (i) [tex]\( f(3) \)[/tex]
To find [tex]\( f(3) \)[/tex], we substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( f(x) = 3x + 5 \)[/tex].
[tex]\[ f(3) = 3(3) + 5 = 9 + 5 = 14 \][/tex]
So, [tex]\( f(3) = 14 \)[/tex].
#### (ii) [tex]\( g(x-3) \)[/tex]
To find [tex]\( g(x-3) \)[/tex], we substitute [tex]\( x-3 \)[/tex] into the function [tex]\( g(x) = 7 - 2x \)[/tex].
[tex]\[ g(x-3) = 7 - 2(x-3) = 7 - 2x + 6 = 13 - 2x \][/tex]
So, [tex]\( g(x-3) = 13 - 2x \)[/tex].
### (b) Find the inverse function [tex]\( g^{-1}(x) \)[/tex]
To find the inverse function of [tex]\( g(x) = 7 - 2x \)[/tex], follow these steps:
1. Replace [tex]\( g(x) \)[/tex] with [tex]\( y \)[/tex]: [tex]\( y = 7 - 2x \)[/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 7 - 2x \\ 2x = 7 - y \\ x = \frac{7 - y}{2} \][/tex]
3. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function:
[tex]\[ g^{-1}(x) = \frac{7 - x}{2} \][/tex]
So, the inverse function [tex]\( g^{-1}(x) = \frac{7 - x}{2} \)[/tex].
### (c) Find [tex]\( h(f(x)) \)[/tex] in the simplest form
To find [tex]\( h(f(x)) \)[/tex], substitute [tex]\( f(x) = 3x + 5 \)[/tex] into the function [tex]\( h(x) = x^2 - 8 \)[/tex].
[tex]\[ h(f(x)) = h(3x + 5) = (3x + 5)^2 - 8 \][/tex]
Expand [tex]\( (3x + 5)^2 \)[/tex]:
[tex]\[ (3x + 5)^2 = 9x^2 + 30x + 25 \][/tex]
So,
[tex]\[ h(f(x)) = 9x^2 + 30x + 25 - 8 = 9x^2 + 30x + 17 \][/tex]
Thus, [tex]\( h(f(x)) = 9x^2 + 30x + 17 \)[/tex].
### (a) Find
#### (i) [tex]\( f(3) \)[/tex]
To find [tex]\( f(3) \)[/tex], we substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( f(x) = 3x + 5 \)[/tex].
[tex]\[ f(3) = 3(3) + 5 = 9 + 5 = 14 \][/tex]
So, [tex]\( f(3) = 14 \)[/tex].
#### (ii) [tex]\( g(x-3) \)[/tex]
To find [tex]\( g(x-3) \)[/tex], we substitute [tex]\( x-3 \)[/tex] into the function [tex]\( g(x) = 7 - 2x \)[/tex].
[tex]\[ g(x-3) = 7 - 2(x-3) = 7 - 2x + 6 = 13 - 2x \][/tex]
So, [tex]\( g(x-3) = 13 - 2x \)[/tex].
### (b) Find the inverse function [tex]\( g^{-1}(x) \)[/tex]
To find the inverse function of [tex]\( g(x) = 7 - 2x \)[/tex], follow these steps:
1. Replace [tex]\( g(x) \)[/tex] with [tex]\( y \)[/tex]: [tex]\( y = 7 - 2x \)[/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 7 - 2x \\ 2x = 7 - y \\ x = \frac{7 - y}{2} \][/tex]
3. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function:
[tex]\[ g^{-1}(x) = \frac{7 - x}{2} \][/tex]
So, the inverse function [tex]\( g^{-1}(x) = \frac{7 - x}{2} \)[/tex].
### (c) Find [tex]\( h(f(x)) \)[/tex] in the simplest form
To find [tex]\( h(f(x)) \)[/tex], substitute [tex]\( f(x) = 3x + 5 \)[/tex] into the function [tex]\( h(x) = x^2 - 8 \)[/tex].
[tex]\[ h(f(x)) = h(3x + 5) = (3x + 5)^2 - 8 \][/tex]
Expand [tex]\( (3x + 5)^2 \)[/tex]:
[tex]\[ (3x + 5)^2 = 9x^2 + 30x + 25 \][/tex]
So,
[tex]\[ h(f(x)) = 9x^2 + 30x + 25 - 8 = 9x^2 + 30x + 17 \][/tex]
Thus, [tex]\( h(f(x)) = 9x^2 + 30x + 17 \)[/tex].
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.