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Practice Problem #1

Verify that f (x) = 2x + 3 and g(x) = the fraction with numerator x minus 3 and denominator 2 are inverses

First, we need to show that f (g(x)) = x.

f(g(x)) = 2 the fraction numerator open paren blank close paren and denominator open paren blank close paren + __

= (__) + __

= __

Answer:

f(g(x)) = 2 (the fraction with numerator x minus 3 and denominator 2) + 3

= (x - 3) + 3

= x

Next, we need to show that g (f(x)) = x

g(f(x)) = the fraction with numerator open paren blank close paren minus 3 and denominator 2

= blank over 2

= __

Answer:

g(f(x)) =the fraction with numerator open paren 2 x plus 3 close paren minus 3 and denominator 2 cancel out the 3s - Because both f (g(x)) and g(f(x)) = x, we can conclude that f(x) and g(x) are inverses.

= 2 x over 2cancel out the 2s

= x

Practice Problems #2

Find the inverse of f(x) = 6 + 5x2. Explain each step and verify that it is an inverse.

f(x) = 6 + 5x - Replace f(x) with y.

y = 6 + 5x2

__ = 6 + 5__2 - Swap x and y - definition of an inverse

Answer: x = 6 + 5y2

__ = 5y2 -Subtract 6 from both sides - subtraction property of equality

Answer: x - 6 = 5y2

the fraction with numerator x minus 6 and denominator blank equals y squared

Divide both sides by 5 - division property of equality

Answer:

the fraction with numerator x minus 6 and denominator 5 equals y squared

plus or minus the square root of the fraction with numerator x minus 6 and denominator 5 equals the square root of y squared -Take the square root of both sides - inverse operation

plus or minus the square root of the fraction with numerator x minus 6 and denominator 5 equals y equals f inverse of x

Verify that this is an inverse of f(x) = 6 + 5x2 and f-1(x) = plus or minus the square root of the fraction with numerator x minus 6 and denominator 5. Find f(f-1(x)).

f of f inverse of x equals 6 plus 5 times open paren plus or minus the square root of the fraction with numerator x minus 6 and denominator 5 close paren squared Substitute f -1(x) for x

f of f inverse of x equals 6 plus 5 times open paren the fraction with numerator x minus 6 and denominator 5 close paren Apply the exponent

f(f-1(x)) = 6 + __ - __ = x

Answer:

f(f-1(x)) = 6 + x - 6 = x

f(f-1(x)) = __

Answer:

f(f-1(x)) = x

f of f inverse of x equals plus or minus the square root of x squared equals x

Is this verified?

Answer: Yes

Verify that this is an inverse of f(x) = 6 + 5x2 and f-1(x) = plus or minus the square root of the fraction with numerator x minus 6 and denominator 5. Find f-1(f(x)).

f inverse of f of x equals plus or minus the square root of the fraction with numerator open paren 6 plus 5 x squared close paren minus 6 and denominator 5
f inverse of f of x equals plus or minus the square root of 5 x squared over 5
f inverse of f of x equals plus or minus the square root of x squared

f-1(f(x)) = ±x

Is this verified?

Answer: Yes

Practice Problem #3

Find the inversion of f of x equals plus or minus the square root of x plus 4 minus 2. Explain each astep and verify that it is an inverse.

f of x equals plus or minus the square root of x plus 4 minus 2

y equals plus or minus the square root of x plus 4 minus 2 Replace f(x) with y

blank equals plus or minus the square root of blank plus 4 minus 2 Swap x and y - definition of an inverse

Answer:

x equals plus or minus the square root of y plus 4 minus 2

blank equals plus or minus the square root of y plus 4 Add 2 to both sides - additive property of equality

Answer: x plus 2 equals plus or minus the square root of y plus 4

open paren x plus 2 close paren squared equals open paren plus or minus the square root of y plus 4 close paren squared square both sides - inverse operation

(x + 2)2 = y + 4

(x + 2)2 __ = __

Answer:

(x + 2)2 - 4 = y

Find the inversion of f of x equals plus or minus the square root of x plus 4 minus 2 and (x + 2)2 - 4 = f -1(x). Find f(f-1(x)) and and f-1 (f(x))

f of f inverse of x equals the square root of open paren plus or minus open paren x plus 2 close paren squared minus 4 close paren plus 4 minus 2 Substitute f -1(x) for x

f of f inverse of x equals the square root of plus or minus open paren x plus 2 close paren squared minus 2 Combine like terms

f(f-1(x)) = (___) - 2

Answer: x+2

f(f-1(x)) = __

Answer: x

Is this verified? __

Answer: Yes

Find f-1 (f(x))

f inverse of f of x equals open paren plus or minus the square root of x plus 4 minus 2 plus 2 close paren squared minus 4 Substitute f -1(x) for x

f inverse of f of x equals open paren plus or minus the square root of x plus 4 close paren squared minus 4 combine like terms

f(f -1(x)) = (__) - 4

Answer: x + 4

f(f -1(x)) = __

Answer: x

Is this verified?___

Answer: yes