"Solutions to Exercises Involving Parent Functions"

           

   A parent function notation is a way in which a function can be represented using symbols and signs as written below.

 

    It is a simpler way of describing a function without lengthy written explanation. In the function g(x), g is written in terms of x. Functions are some times represented using the letter y. That is y=f(x) or y=g(x). 

   To solve an exercise in functions simply means producing another function or coming up with a numerical value. The solutions to the exercises below will see the production of new functions.

 

1)      The function g is related one of the parent functions g(x) = x²+6. The parent function is f(x)= x². Use function notation to write g in terms of f.

                Solution.

g(x)=x²+6 , f(x) =x²

g in terms of f is g(f(x)). In place of x in g(x) put f(x). This is also known as the composite function g of f.

g(f(x))= g(x²)

g(f(x))=(x²)²+6

g(f(x))=x⁴+6.

We can write it  as y= g(f(x))

2)      The function g is related to one of the parent functions g(x)= x²-3. The parent function is  f(x)=x².

Solution

g(x)= x²-3 , f(x)=x²

g in terms of f is g(f(x)). In place of x in g(x) put f(x). Also known as the composite function g of f

g(f(x))= g(x²)

g(f(x))= (x²)²-3

g(f(x))=x⁴-3.

We can write it as y=g(f(x))

3)      The function g is related to one parent function, g(x)= -(x-2)².

a)      Identify the parent function f.

b)      Use function notation to write g in terms of f.

    Solution

A  parent function is the simplest form of a family of a family of functions that preserves the definition or shape of the entire family. Examples here linear function (y=x), quadratic function (y=x²), polynomial functions, exponential functions(y=2^x), logarithmic functions (y=log x), absolute value functions( y=Ix-2I ) and functions with root symbol . All other functions are derived from each of these parent functions.

 To identify a parent function, remove all the arithmetic that is subtraction, addition, multiplication, division and leave one higher operation on just x. The higher operation is  root, absolute value, or exponent.

a)      g(x)= -(x-2)² (remove – sign and -2 from the right hand side)

g(x)=(x)²

g(x)= x² ( the parent function). This is a quadratic function

b)      g(f(x))= -(f(x)-2)² g written in terms of f

g(f(x))=-(x²-2)²  or y=g(f(x)) g written in terms of f

4)      The function g is related to one of the parent functions where g(x)= Ix-1I+5

a)      Identify the parent function f.

b)      Use function notation to write g in terms of f.

Solution

a)      g(x) = I x-1I+5 ( remove -1, +5)

f(x)=Ix I ( the parent function). This is an absolute value function.

b)      g(f(x))= I f(x)-1 I +5

g(f(x))= I IxI-1I +5 or  y=g(f(x)). g written in terms of f.

5) The function g is related to one parent function or g(x)= 2x^0.5.

                a) Identify the parent function f

                b) Use function notation to write g in terms of f

                                   solution

a)        y=2x^{0. 5} ( strip 2 from the right hand side)

  or y = x^{0. 5} ( the parent function) . This is a radical function.

b)      g(f(x))

g(f(x))

g(f(x))  , y=g(f(x))

 

Finally, to write a function like g (x) in terms of f(x), simply replace the variable x in g by the function f(x). Rewrite your function now as y=? Remember, when we replace x in g by f(x), f(x) becomes the new variable of g, that is g(f(x)). The new function will be y=g(f(x)). 

 To identify a parent function, strip away all the arithmetic in the given function to leave behind one higher order operation on just x, for example exponent, absolute value, radical.

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