Finding Inverses
How do we prove that two functions are inverses algebraically? (not graphically)
f(g(x))=g(f(x))=x
The equation of an exponential function is in the form f(x)=ab^x. What point on the graph does a represent?
y-intercept
Find log_4(256) .
4
Expand log_4(xz) .
log_4(x)+log_4(z)
Find the inverse of f(x)=\frac{5x+6}{4} .
f^-1(x)=\frac(4x-6)(5)
The equation of an exponential function is in the form f(x)=ab^x. Draw sketches describing what the following would look like graphically:
a. b>1
b. 0<b<1
c. b<0
a. increasing exponential (Growth)
b. decreasing exponential (Decay)
c. not possible
Find log(10) .
1
Condense log_5(a)-log_5(9) .
log_5(\frac(a)(9))
Find the inverse of g(x)=4x^3-1 .
g^-1(x)=\root(3)(\frac(x+1)(4))
An exponential function is known to have a y-intercept at (0,5) and another coordinate at (1,8). Find the equation of the function in the form y=ab^x .
y=5(1.6)^x
Rewrite 2^(x+1)=z as a logarithm.
log_2(z)=x+1
Expand log_3(3x^2) .
1+2log_3(x)
Find the inverse of f(x)=log_2(x-3)+9 .
f^-1(x)=2^(x-9)+3
Walter White is performing experiments on different chemicals compounds. He finds that the mass of a certain compound decreases by 12% every hour after it is exposed to air. How much of a 52-gram sample of this chemical compound will remain after 6 hours? Round to the nearest gram.
24 grams
Rewrite log_(y+1)(x+9)=z as an exponential.
(y+1)^z=x+9
Condense 4log_2(y)-3log_2(z)+log_2(w) .
log_2(\frac(y^4w)(z^3))
Find the inverse of g(x)=3(4)^(x+1) .
g^-1(x)=log_4(\frac(x)(3))-1
A certain type of bamboo is known to grow in height exponentially. If the bamboo is at a height of 3.3 feet when it is first planted, and it increases in height by 6% each week, how many weeks will it take for the bamboo to reach a height of 10 feet? Round to the nearest week.
19 weeks
Solve the following equation. Leave as an exact answer, not a decimal approximation.
4*5^x=560
x=log_5(140)
Expand log_4((2x^2)/y^3) .
1/2+2log_4(x)-3log_4(y)
Simplify the following: (5^(log_3(81))-log_2(1))^(1/2)
25