Answer :
Certainly! Here is the detailed step-by-step solution for each part of the given problem:
Given the functions:
[tex]\[ g(x) = -3x + 1 \][/tex]
[tex]\[ f(x) = x^2 + 7 \][/tex]
[tex]\[ h(x) = \frac{12}{x} \][/tex]
[tex]\[ j(x) = 2x + 9 \][/tex]
### Part 1: Evaluate the following expressions
a. [tex]\( g(10) = \)[/tex]
To find [tex]\( g(10) \)[/tex], substitute [tex]\( x = 10 \)[/tex] into the function [tex]\( g(x) \)[/tex].
[tex]\[ g(10) = -3(10) + 1 = -30 + 1 = -29 \][/tex]
b. [tex]\( f(3) = \)[/tex]
To find [tex]\( f(3) \)[/tex], substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( f(x) \)[/tex].
[tex]\[ f(3) = 3^2 + 7 = 9 + 7 = 16 \][/tex]
c. [tex]\( h(-2) = \)[/tex]
To find [tex]\( h(-2) \)[/tex], substitute [tex]\( x = -2 \)[/tex] into the function [tex]\( h(x) \)[/tex].
[tex]\[ h(-2) = \frac{12}{-2} = -6 \][/tex]
d. [tex]\( j(7) = \)[/tex]
To find [tex]\( j(7) \)[/tex], substitute [tex]\( x = 7 \)[/tex] into the function [tex]\( j(x) \)[/tex].
[tex]\[ j(7) = 2(7) + 9 = 14 + 9 = 23 \][/tex]
e. [tex]\( h(a) = \)[/tex]
Assume [tex]\( a = 5 \)[/tex] for example purposes. To find [tex]\( h(a) \)[/tex], substitute [tex]\( x = 5 \)[/tex] into the function [tex]\( h(x) \)[/tex].
[tex]\[ h(5) = \frac{12}{5} = 2.4 \][/tex]
f. [tex]\( g(b + c) = \)[/tex]
Assume [tex]\( b = 2 \)[/tex] and [tex]\( c = 3 \)[/tex] for example purposes. To find [tex]\( g(b + c) \)[/tex], first calculate [tex]\( b + c \)[/tex] and then substitute into [tex]\( g(x) \)[/tex].
[tex]\[ b + c = 2 + 3 = 5 \][/tex]
[tex]\[ g(b + c) = g(5) = -3(5) + 1 = -15 + 1 = -14 \][/tex]
g. [tex]\( f(h(x)) = \)[/tex]
Assume [tex]\( x = 4 \)[/tex] for example purposes. To find [tex]\( f(h(x)) \)[/tex], first calculate [tex]\( h(x) \)[/tex] and then substitute the result into [tex]\( f(x) \)[/tex].
[tex]\[ h(4) = \frac{12}{4} = 3 \][/tex]
[tex]\[ f(h(4)) = f(3) = 3^2 + 7 = 9 + 7 = 16 \][/tex]
h. Find [tex]\( x \)[/tex] if [tex]\( g(x) = 16 \)[/tex]
To find [tex]\( x \)[/tex] when [tex]\( g(x) = 16 \)[/tex]:
[tex]\[ -3x + 1 = 16 \][/tex]
[tex]\[ -3x = 16 - 1 \][/tex]
[tex]\[ -3x = 15 \][/tex]
[tex]\[ x = \frac{15}{-3} = -5 \][/tex]
i. Find [tex]\( x \)[/tex] if [tex]\( h(x) = -2 \)[/tex]
To find [tex]\( x \)[/tex] when [tex]\( h(x) = -2 \)[/tex]:
[tex]\[ \frac{12}{x} = -2 \][/tex]
[tex]\[ 12 = -2x \][/tex]
[tex]\[ x = \frac{12}{-2} = -6 \][/tex]
### Part 2: Translate the following statements into coordinate points
j. Find [tex]\( x \)[/tex] if [tex]\( f(x) = 23 \)[/tex]
To find [tex]\( x \)[/tex] when [tex]\( f(x) = 23 \)[/tex]:
[tex]\[ x^2 + 7 = 23 \][/tex]
[tex]\[ x^2 = 23 - 7 \][/tex]
[tex]\[ x^2 = 16 \][/tex]
[tex]\[ x = \pm 4 \][/tex]
So the points are [tex]\((4, 23)\)[/tex] and [tex]\((-4, 23)\)[/tex].
a. [tex]\( f(-1) = 1 \)[/tex]
This translates to the coordinate point [tex]\((-1, 1)\)[/tex].
Given the functions:
[tex]\[ g(x) = -3x + 1 \][/tex]
[tex]\[ f(x) = x^2 + 7 \][/tex]
[tex]\[ h(x) = \frac{12}{x} \][/tex]
[tex]\[ j(x) = 2x + 9 \][/tex]
### Part 1: Evaluate the following expressions
a. [tex]\( g(10) = \)[/tex]
To find [tex]\( g(10) \)[/tex], substitute [tex]\( x = 10 \)[/tex] into the function [tex]\( g(x) \)[/tex].
[tex]\[ g(10) = -3(10) + 1 = -30 + 1 = -29 \][/tex]
b. [tex]\( f(3) = \)[/tex]
To find [tex]\( f(3) \)[/tex], substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( f(x) \)[/tex].
[tex]\[ f(3) = 3^2 + 7 = 9 + 7 = 16 \][/tex]
c. [tex]\( h(-2) = \)[/tex]
To find [tex]\( h(-2) \)[/tex], substitute [tex]\( x = -2 \)[/tex] into the function [tex]\( h(x) \)[/tex].
[tex]\[ h(-2) = \frac{12}{-2} = -6 \][/tex]
d. [tex]\( j(7) = \)[/tex]
To find [tex]\( j(7) \)[/tex], substitute [tex]\( x = 7 \)[/tex] into the function [tex]\( j(x) \)[/tex].
[tex]\[ j(7) = 2(7) + 9 = 14 + 9 = 23 \][/tex]
e. [tex]\( h(a) = \)[/tex]
Assume [tex]\( a = 5 \)[/tex] for example purposes. To find [tex]\( h(a) \)[/tex], substitute [tex]\( x = 5 \)[/tex] into the function [tex]\( h(x) \)[/tex].
[tex]\[ h(5) = \frac{12}{5} = 2.4 \][/tex]
f. [tex]\( g(b + c) = \)[/tex]
Assume [tex]\( b = 2 \)[/tex] and [tex]\( c = 3 \)[/tex] for example purposes. To find [tex]\( g(b + c) \)[/tex], first calculate [tex]\( b + c \)[/tex] and then substitute into [tex]\( g(x) \)[/tex].
[tex]\[ b + c = 2 + 3 = 5 \][/tex]
[tex]\[ g(b + c) = g(5) = -3(5) + 1 = -15 + 1 = -14 \][/tex]
g. [tex]\( f(h(x)) = \)[/tex]
Assume [tex]\( x = 4 \)[/tex] for example purposes. To find [tex]\( f(h(x)) \)[/tex], first calculate [tex]\( h(x) \)[/tex] and then substitute the result into [tex]\( f(x) \)[/tex].
[tex]\[ h(4) = \frac{12}{4} = 3 \][/tex]
[tex]\[ f(h(4)) = f(3) = 3^2 + 7 = 9 + 7 = 16 \][/tex]
h. Find [tex]\( x \)[/tex] if [tex]\( g(x) = 16 \)[/tex]
To find [tex]\( x \)[/tex] when [tex]\( g(x) = 16 \)[/tex]:
[tex]\[ -3x + 1 = 16 \][/tex]
[tex]\[ -3x = 16 - 1 \][/tex]
[tex]\[ -3x = 15 \][/tex]
[tex]\[ x = \frac{15}{-3} = -5 \][/tex]
i. Find [tex]\( x \)[/tex] if [tex]\( h(x) = -2 \)[/tex]
To find [tex]\( x \)[/tex] when [tex]\( h(x) = -2 \)[/tex]:
[tex]\[ \frac{12}{x} = -2 \][/tex]
[tex]\[ 12 = -2x \][/tex]
[tex]\[ x = \frac{12}{-2} = -6 \][/tex]
### Part 2: Translate the following statements into coordinate points
j. Find [tex]\( x \)[/tex] if [tex]\( f(x) = 23 \)[/tex]
To find [tex]\( x \)[/tex] when [tex]\( f(x) = 23 \)[/tex]:
[tex]\[ x^2 + 7 = 23 \][/tex]
[tex]\[ x^2 = 23 - 7 \][/tex]
[tex]\[ x^2 = 16 \][/tex]
[tex]\[ x = \pm 4 \][/tex]
So the points are [tex]\((4, 23)\)[/tex] and [tex]\((-4, 23)\)[/tex].
a. [tex]\( f(-1) = 1 \)[/tex]
This translates to the coordinate point [tex]\((-1, 1)\)[/tex].
