Answer to Question #273513 in Linear Algebra for Zaia grgs

Question #273513

For each of the following functions determine the inverse image of T = {x ∈ R : 0 ≤ x 2 − 25}. 

1. f : R → R defined by f(x) = 3x3.

2. g : R + → R defined by g(x) = ln(x).

3. h : R → R defined by h(x) = x − 9.

 


1
Expert's answer
2021-11-30T18:27:04-0500

1.

a. Write y=f(x)y=f(x)


y=3x3.y = 3x^3.

b. Interchange xx and yy


x=3y3x=3y^3

c. Solve for yy


y=(x/3)1/3y=(x/3)^{1/3}

d. Write f1(x)=yf^{-1}(x)=y


f1(x)=(x/3)1/3f^{-1}(x)=(x/3)^{1/3}

0x225=>5x50\leq x^2\leq25=>-5\leq x\leq5

f1(T)=[(5/3)1/3,(5/3)1/3]f^{-1}(T)=[-(5/3)^{1/3}, (5/3)^{1/3}]

2.

a. Write y=g(x)y=g(x)


y=lnxy = \ln x

b. Interchange xx and yy


x=lnyx=\ln y

c. Solve for yy


y=exy=e^x

d. Write g1(x)=yg^{-1}(x)=y


g1(x)=ex,x>0,xRg^{-1}(x)=e^x,x>0, x\in \R

0x225,x>0=>0<x50\leq x^2\leq25, x>0=>0<x\leq5

g1(T)=[1,e5]g^{-1}(T)=[1, e^5]

3.

a. Write y=h(x)y=h(x)


y=x9.y = x-9.

b. Interchange xx and yy


x=y9x=y-9

c. Solve for yy


y=x+9y=x+9

d. Write h1(x)=yh^{-1}(x)=y


h1(x)=x+9h^{-1}(x)=x+9

0x225=>5x50\leq x^2\leq25=>-5\leq x\leq5

h1(T)=[4,14]h^{-1}(T)=[4, 14]


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