5.1 Inverse Functions
Find the inverse of the relation.
{(0, 5), (-7, 7), (-5, 9), (5, -5)}
{(5,0), (7,-7), (9,-5), (-5,5)
Use a calculator to find the value to the nearest ten-thousandth:
e^-2.45
approx 0.0863
Graph the equation on paper:
y=log(base 7) x
The logarithmic equation y=log(base 7) x is equivalent to the exponential equation x=7^y.
Plug in 0, -1, -2, 1, 2 for y-values into x=7^y
(1,0), (1/7,-1), (1/49,-2), (7,1), (49,2)
Plot the points and connect the dots with a smooth curve
Express as the sum or difference of logarithms:
log(base 5)21x
log(base 5)21+log(base 5)x
Solve:
3^3x=9
x=2/3
Graph the inverse of x=y^2+6. Reflect the graph across the line y=x to obtain the graph of its inverse relation.
Plug y-values -2, -1, 0, 1, and 2 into x=y^2+6 to get x-values
X Y
10 -2
7 -1
6 0
7 1
10 2
Then flip the ordered pairs around to graph the inverse function across the y=x line.
(-2,10), (-1,7), (0,6), (1,7), (2,10)
Use the transformation to graph f(x)=-3^x-4; parent function is f(x)=3^x
Plug in -2, -1, 0, 1, 2 for x-values into f(x)=3^x
x y=3^x
-2 1/9
-1 1/3
0 1
1 3
2 9
Plot the points
To get -3^x the graph should be reflected over the x-axis
Final part of the function is the shift the graph of the function -3^x, 4 units down
Find the logarithm:
log(base 10)1,000,000
Let the expression equal 'w'
Now you have a logarithmic equation. To convert the equation, use the definition of logarithms:
(y = log(base a) x) is equivalent to a^y=x
In this case, a=10 and y=w
10^w=1,000,000
To find 'w', write 1,000,000 as a power of 10: 10^6
10^w=10^6
Base cancels out so w=6
Express as a product:
ln (7 squareroots of 4)
1/7ln4
Solve for x.
log(base 2)x = 4
x=16
Determine whether the following function is one-to-one using the definition of a one-to-one function:
f(x)=1/5x-8
A function f is one-to-one if different inputs have different outputs that is, if a≠b, then f(a)≠f(b). Or, if when the outputs are the same, the inputs are the same that is, if f(a)=f(b), then a = b. Use either of these conditions to determine whether f is one-to-one or not.
Plug f(a) into the equation: f(a)=1/5a-8
Plug f(b) into the equation: f(b)=1/5b-8,
Now set the 2 expressions equal to each other: 1/5a-8 = 1/5b-8
Simplify by adding 8 to both sides: 1/5a = 1/5b
Rationalize the expression by multiplying both sides by the denominator 5 to get rid of it: a = b
So, for the given function, if f(a) = f(b), then a = b is true.
Therefore, the given function is a one-to-one function.
Graph f(x)=e^x+2
Plug in 0, 1, 2, -1 for x-values
x f(x)=e^x+2
0 3.0
1 4.7
2 9.4
-1 2.4
Then plot the ordered pairs
Find the logarithm:
log(base3) 1/81
-4
Express as a difference of logarithms:
log(base b)Z/17
log(base b)Z - log(base b)17
Solve the exponential equation:
3^2x+1 = 4^x
x = approx. -1.355
Graph the function and its inverse using a graphing calculator:
f(x)=0.4x+2.5
To find the inverse of the function replace f(x) with y: y=0.4x+2.5
Now interchange x and y: x=0.4y+2.5
Solve for y by subtracting 2.5 from both sides: 0.4y=x - 2.5
Now divide by sides by 0.4 to isolate y: y=2.5x - 6.25
Finally replace y with f^-1(x): f^-1(x)=2.5x-6.25
Graph the function: f(x)=3/4e^x
Plug in -2, -1, 0, 1, 2 for x-values
x f(x)=3/4e^x
-2 approx. 0.102
-1 approx. 0.276
0 approx. 0.75
1 approx. 2.039
2 approx. 5.542
Plot the points and draw the graph
Find the following without using a calculator:
ln(15squareroot) of e
1/15
Express log(base b)p^5q^8/m^6b^4 in terms of sums and differences of logarithms.
5log(base b)p + 8log(base b)q - 6log(base b)m - 4
Solve the following logarithmic equation:
log(base 125)1/5 = x
-1/3
For the following function, a) determine whether it is one-to-one; b) if it is one-to-one, find its inverse function:
f(x)=x−7
a) The function is one-to-one because it passes the horizontal line test
b) f^-1(x)=x+7
Graph the following piecewise function:
{e^-x-5, for x<-2}
f(x)= {x+2, for -2<(or equal to)x<1}
{x^2, for x>(or equal to)1}
Plug in -2.25, -2.50, -2.75 for x-values into f(x)=e^-x-5
(-2.25,4.488), (-2.50,7.182), (-2.75,10.643)
Graph this piece of the function for x<-2
Plug in -2, -1, 0, 1 for x-values into f(x)=x+2
(-2,0), (-1,1), (0,2), (1,3)
Graph this piece of the function for -2<(or equal to)x<1
Plug in 1, 3, 4 for x-values into f(x)=x^2
(1,4), (3,9), (4,16)
Graph this piece of the function for x>(or equal to)1
Find the logarithm using natural logarithms and the change-of-base formula:
log(base700)37
approx. 0.5512
Given that log(base b)(5)≈1.609, log(base b)(11)≈2.398, and log(base b)(16)≈2.773, find the logarithm of log(base b)1/55
-4.007
Solve the following equation:
8^x-4 = 64(3^x)
x= approx. 12.721