Question

10 10 f(x) g(x) a. What is the domain of f(x)? b. What is the range of f(x)? c. What is (gof)(-5)? d. Solve f(x) = 10 for x. e. Bonus: Solve f(x) = g(x)

Answer 1

a)domain is all values of x through which graph passes.

f(x) takes all value from left side (negative infinity - arrow shows that) upto x=6 . so

$\LARGE \boldsymbol{domain=(-\infty,6]}$

b) range is all values of y through which graph passes . graph passes through y=-20 to y=30 . so

range = [-20, 30

c) we know  $\large (f \circ g)(x)=f(g(x))$

so $\large (g\circ f)(-5)=g(f(-5))$

from graph of f , f(-5) is 20 . so

$\large (g\circ f)(-5)=g(20)$

now we need to find g(20) .

g(20) is not in graph , we can find this by finding equation for line on right side in graph g .

standard form of a line is f(x)=mx+c . here m is slope and c is y intercept .

it passes throgh (4,-10) and (6,0) . so slope of this line is

$\large slope=\frac{0-^-10}{6-4}=5$

here y intercept is -30 . so

g(x)= 5x-30 .

so g(20)= 5*20-30=70

$\large (g\circ f)(-5)=g(20)=70$

so answer is

$\large \boldsymbol{(g\circ f)(-5)=70}$

d)to solve this draw a line y=10 . then find point of intersection of line with graph . then x cordinate of this points are solution.

y=10 touches graph at (-6,10) ,(0,10) and (6,10).

f(x) is 10 for three value of x . they are -6,0 and 6

e) to solve this , find x values which make f(x) and g(x) are same.

from graph f(-6)=10 . g(-6)=10

so for x=-6 , f(x)=g(x) . so this is solution

so answer is x=-6