# Find the Exponential Function Given a Point (-3,-2) (-3,-2)
To find an exponential function, f(x)=ax, containing the point, set f(x) in the function to the y value -2 of the point, and set x to the x value -3 of the point.
-2=a-3
Solve the equation for a.
Rewrite the equation as a-3=-2.
a-3=-2
Rewrite the expression using the negative exponent rule b-n=1bn.
1a3=-2
Solve for a.
Multiply each term by a3 and simplify.
Multiply each term in 1a3=-2 by a3.
1a3⋅a3=-2⋅a3
Cancel the common factor of a3.
Cancel the common factor.
1a3⋅a3=-2⋅a3
Rewrite the expression.
1=-2⋅a3
1=-2a3
1=-2a3
Rewrite the equation as -2a3=1.
-2a3=1
Move 1 to the left side of the equation by subtracting it from both sides.
-2a3-1=0
Factor -1 out of -2a3-1.
Factor -1 out of -2a3.
-(2a3)-1=0
Rewrite -1 as -1(1).
-(2a3)-1⋅1=0
Factor -1 out of -(2a3)-1(1).
-(2a3+1)=0
-(2a3+1)=0
Multiply each term in -(2a3+1)=0 by -1
Multiply each term in -(2a3+1)=0 by -1.
-(2a3+1)⋅-1=0⋅-1
Simplify -(2a3+1)⋅-1.
Apply the distributive property.
(-(2a3)-1⋅1)⋅-1=0⋅-1
Multiply.
Multiply 2 by -1.
(-2a3-1⋅1)⋅-1=0⋅-1
Multiply -1 by 1.
(-2a3-1)⋅-1=0⋅-1
(-2a3-1)⋅-1=0⋅-1
Apply the distributive property.
-2a3⋅-1-1⋅-1=0⋅-1
Multiply.
Multiply -1 by -2.
2a3-1⋅-1=0⋅-1
Multiply -1 by -1.
2a3+1=0⋅-1
2a3+1=0⋅-1
2a3+1=0⋅-1
Multiply 0 by -1.
2a3+1=0
2a3+1=0
Subtract 1 from both sides of the equation.
2a3=-1
Divide each term by 2 and simplify.
Divide each term in 2a3=-1 by 2.
2a32=-12
Cancel the common factor of 2.
Cancel the common factor.
2a32=-12
Divide a3 by 1.
a3=-12
a3=-12
Move the negative in front of the fraction.
a3=-12
a3=-12
Take the cube root of both sides of the equation to eliminate the exponent on the left side.
a=-123
Simplify -123.
Rewrite -12 as ((-1)3)312.
Rewrite -1 as (-1)3.
a=(-1)3123
Rewrite -1 as (-1)3.
a=((-1)3)3123
a=((-1)3)3123
Pull terms out from under the radical.
a=(-1)3123
Raise -1 to the power of 3.
a=-123
Rewrite 123 as 1323.
a=-1323
Any root of 1 is 1.
a=-123
Multiply 123 by 232232.
a=-(123⋅232232)
Combine and simplify the denominator.
Multiply 123 and 232232.
a=-23223232
Raise 23 to the power of 1.
a=-232231232
Use the power rule aman=am+n to combine exponents.
a=-232231+2
Add 1 and 2.
a=-232233
Rewrite 233 as 2.
Use axn=axn to rewrite 23 as 213.
a=-232(213)3
Apply the power rule and multiply exponents, (am)n=amn.
a=-232213⋅3
Combine 13 and 3.
a=-232233
Cancel the common factor of 3.
Cancel the common factor.
a=-232233
Divide 1 by 1.
a=-23221
a=-23221
Evaluate the exponent.
a=-2322
a=-2322
a=-2322
Simplify the numerator.
Rewrite 232 as (22)13.
a=-2232
Raise 2 to the power of 2.
a=-432
a=-432
a=-432
a=-432
a=-432
Substitute each value for a back into the function f(x)=ax to find each possible exponential function.
f(x)=(-432)x
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