For the wuadratic function in standard form

Question

identify the damain and range : f(x)=(1/4)(x-(3/4))^2 = (1/2)?

Answer 1

f(x) = (1/4)[x-(3/4)]² = 1/2

(1/4)[x-(3/4)]² = 1/2  and f(x) = 1/2

Multiply each side by 4.

(4)(1/4)[x-(3/4)]² = (1/2)(4)

Simplify

[x-(3/4)]² = 2

Recall: [ (a - b)² = a² - 2ab + b² ]

[x² - 2(x)(3/4) + (3/4)²] = 2

[x² - (x)(3/2) + (9/16)] = 2

Here LCm of 1, 2, 16 is 16

(16x² - 24x + 9)/(16)= 1

Multiply each side by 16.

[(16x² - 24x + 9)/(16)](16) = 1(16)

16x² - 24x + 9 = 16

Subtract 16 from each side.

16x² - 24x + 9 - 16 = 16 - 16

16x² - 24x - 7 = 0

16x² + 4x - 28x - 7 = 0

Take out common terms.

4x(4x + 1) -7(4x + 1) = 0

Take out common factors.

(4x + 1)(4x - 7) = 0

(4x + 1) = 0  or (4x - 7) = 0

4x + 1 = 0

Subtract 1 from each side.

4x + 1 - 1 = -1

4x = -1

Divide each side by 24.

4x/4 = -1/4

x = -1/4

And 4x - 7 = 0

Then x = 7/4

Domain is [7/4, -1/4]

Range is 1/2

Answer 2

The function is f(x ) = y = (1/4) (x - 3/4)² - 1/2.

The above function represents a parabola vertex form  y = a (x - h )^2 + k.

Domain: The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function. It is the set of all real numbers for which a function is mathematically defined.

Domain of a function f(x ) is set of those values of x which will make the function mathematically legal or correct..certain operations like division by zero , square root of a negative number do not exist in real maths.

1. Domain excludes x - values that result in division by zero.

2. Domain excludes x - values that result in even roots of negative numbers.

The function f(x ) = (1/4) (x - 3/4)² - 1/2 is a parabola function.There are no rational or radical expressions, so there is nothing that will restrict the domain. Any real number can be used for x to get a meaningful output.

Therefore domain of f(x ) = (1/4) (x - 3/4)² - 1/2 is all real numbers.

Range: The range is the set of all possible output values (usually the variable y, or sometimes expressed as f(x)), which result from using a particular function.

The function f(x) = (1/4) (x - 3/4)² - 1/2 represents a parabola vertex form  y = a (x - h )^2 + k .

 a  = 1/4 , h  = 3/4 and k  = -1/2.

a  is positive number the parabola opens up and has minimum value.

When the parabola opens up it has a minimum point which is the vertex of parabola (3/4, -1/2)

In the minimum point y  = k = -1/2.  So the graph of parabola cannot be lower than -1/2.

Thus the range of function y  ≥ -1/2.

Domain of function is all real numbers.

Range of the function is  {y |y  ≥ -1/2}.