Which Equation is the Inverse of Y 16×2 1
Which Equation is the Inverse of Y 16×2 1
An
inverse function
or an anti role is defined as a function, which can reverse into some other function. In simple words, if any office “f” takes x to y then, the inverse of “f” will have y to x. If the function is denoted by ‘f’ or ‘F’, then the inverse function is denoted by f
^{1}
or F
^{1}
. One should not misfile (one) with exponent or reciprocal here.
If f and g are changed functions, so f(x) = y if and but if g(y) = ten
In trigonometry, the
inverse sine function
is used to find the measure of angle for which sine part generated the value. For example, sin
^{i}
(1) = sin^{i}
(sin 90) = 90 degrees. Hence, sin 90 degrees is equal to 1.
Tabular array of Contents:
 Definition
 Graph
 How to Find Inverse Function
 Types
 Finding Changed Function Using Algebra
 Example
Definition
A function accepts values, performs particular operations on these values and generates an output. The changed function agrees with the resultant, operates and reaches dorsum to the original function.
The inverse function returns the original value for which a function gave the output.
If you consider functions, f and g are inverse, f(g(x)) = g(f(x)) = ten. A role that consists of its changed fetches the original value.
Example: f(x) = 2x + 5 = y
Then, g(y) = (yfive)/2 = x is the inverse of f(x).
Note:
 The relation, developed when the contained variable is interchanged with the variable which is dependent on a specified equation and this inverse may or may not be a role.
 If the inverse of a function is itself, then it is known as inverse role, denoted past f^{ane}(x).
Inverse Function Graph
The graph of the changed of a function reflects ii things, 1 is the part and second is the inverse of the office, over the line y = x. This line in the graph passes through the origin and has slope value i. It tin can be represented as;
y = f^{ane}(10)
which is equal to;
x = f(y)
This relation is somewhat similar to y = f(x), which defines the graph of f but the part of x and y are reversed here. So if we have to describe the graph of f^{1}, then we have to switch the positions of ten and y in axes.
Video Lesson
Changed Functions
How to Notice the Inverse of a Office?
Mostly, the method of computing an inverse is swapping of coordinates x and y. This newly created inverse is a relation but not necessarily a office.
The original role has to be a itoi function to assure that its inverse will also be a office. A function is said to be a one to one function only if every second element corresponds to the first value (values of 10 and y are used only once).
Yous can apply on the horizontal line test to verify whether a office is a anetoane role. If a horizontal line intersects the original function in a single region, the office is a onetoone function and inverse is also a role.
 Changed Function Formula
 Inverse Trigonometric Functions
 Of import Questions Form 12 Maths Affiliate ii Inverse Trigonometric Functions
Types of Changed Function
There are diverse types of changed functions like the inverse of trigonometric functions, rational functions, hyperbolic functions and log functions. The inverses of some of the nigh mutual functions are given below.
Function  Inverse of the Function  Comment 

+  –  
×  /  Don’t split up by 0 
ane/10  1/y  x and y non equal to 0 
x^{2}  √y  x and y ≥ 0 
x^{n}  y^{1/n}  n is not equal to 0 
eastward^{x}  ln(y)  y > 0 
a^{ten}  log a(y)  y and a > 0 
Sin (x)  Sin^{1} (y) 
– π/2 to + π/2 
Cos (x)  Cos^{1} (y) 
0 to π 
Tan (10)  Tan^{1} (y) 
– π/two to + π/2 
Inverse Trigonometric Functions
The inverse trigonometric functions are also known every bit
arc part
as they produce the length of the arc, which is required to obtain that particular value. There are halfdozen inverse trigonometric functions which include arcsine (sin^{one}), arccosine (cos^{1}), arctangent (tan^{1}), arcsecant (sec^{one}), arccosecant (cosec^{ane}), and arccotangent (cot^{1}).
Inverse Rational Part
A rational function is a function of form f(x) = P(x)/Q(ten) where Q(x) ≠ 0. To find the inverse of a rational function, follow the following steps. An case is also given below which tin can assistance you to understand the concept improve.

Step ane:
Replace f(x) = y 
Step 2:
Interchange x and y 
Step 3:
Solve for y in terms of x 
Step 4:
Replace y with f^{i}(10) and the changed of the office is obtained.
Changed Hyperbolic Functions
Just like inverse trigonometric functions, the inverse hyperbolic functions are the inverses of the hyperbolic functions. At that place are mainly 6 inverse hyperbolic functions be which include sinh^{1}, cosh^{1}, tanh^{1}, csch^{1}, coth^{1}, and sech^{1}. Check out changed hyperbolic functions formula to learn more about these functions in detail.
Inverse Logarithmic Functions and Changed Exponential Role
The natural log functions are changed of the exponential functions. Check the following example to empathize the inverse exponential office and logarithmic function in detail. Also, get more insights of how to solve similar questions and thus, develop problemsolving skills.
Finding Inverse Function Using Algebra
Put “y” for “f(x)” and solve for x:
The function:  f(x)  =  2x+iii 
Put “y” for “f(x)”:  y  =  2x+3 
Subtract 3 from both sides:  ythree  =  2x 
Dissever both sides past 2:  (y3)/2  =  ten 
Swap sides:  ten  =  (yiii)/2 
Solution (put “f^{1}(y)” for “ten”) :  f^{i}(y)  =  (y3)/2 
Changed Functions Example
Example ane:
Find the inverse of the function f(x) = ln(x – 2)
Solution:
Beginning, supervene upon f(x) with y
So, y = ln(x – two)
Replace the equation in exponential mode , ten – 2 = e^{y}
Now, solving for x,
x = two + eastward^{y}
Now, replace 10 with y and thus, f^{one}(ten) = y = ii + e^{y}
Example 2:
Solve: f(ten) = 2x + 3, at ten = 4
Solution:
We accept,
f(4) = 2 × four + 3
f(4) = 11
Now, let’s apply for reverse on 11.
f^{1}(11) = (11 – three) / 2
f^{1}(11) = four
Magically nosotros become 4 again.
Therefore, f^{1}(f(4)) = 4
So, when nosotros apply function f and its reverse f^{i}
gives the original value back again, i.e, f^{1}(f(ten)) = x.
Example 3:
Find the inverse for the function f(x) = (3x+2)/(x1)
Solution:
First, supplant f(10) with y and the office becomes,
y = (3x+two)/(101)
Past replacing x with y we become,
x = (3y+2)/(yi)
Now, solve y in terms of 10 :
ten (y – 1) = 3y + ii
=> xy – x = 3y +2
=> xy – 3y = 2 + x
=> y (10 – 3) = 2 + x
=> y = (two + x) / (x – 3)
So, y = f^{1}(x) = (ten+ii)/(xthree)
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Frequently Asked Questions – FAQs
What is the inverse role?
An changed office is a role that returns the original value for which a function has given the output. If f(ten) is a function which gives output y, then the inverse role of y, i.e. f^{i}(y) will return the value 10.
How to detect the changed of a function?
Suppose, f(ten) = 2x + iii is a part.
Let f(x) = 2x+3 = y
y = 2x+iii
x = (y3)/2 = f^{one}(y)
This is the inverse of f(x).
Are inverse office and reciprocal of function, same?
One should not get confused inverse function with reciprocal of function. The inverse of the function returns the original value, which was used to produce the output and is denoted past f^{1}(10). Whereas reciprocal of part is given past 1/f(x) or f(x)^{1}
For instance, f(x) = 2x = y
f^{one}(y) = y/2 = x, is the changed of f(10).
But, 1/f(10) = one/2x = f(x)^{ane}
is the reciprocal of function f(ten).
What is the inverse of 1/x?
Let f(x) = 1/ten = y
Then inverse of f(x) will be f^{1}(y).
f^{1}(y) = 1/x
How to solve inverse trigonometry function?
If nosotros take to find the inverse of trigonometry function sin x = ½, then the value of 10 is equal to the bending, the sine function of which angle is ½.
As we know, sin 30° = ½.
Therefore, sin x = ½
x = sin^{ane}(½) = sin^{1}
(sin 30°) = 30°
Which Equation is the Inverse of Y 16×2 1
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