Exercise 34 Page 257

We want to graph the given function in a graphing calculator. To do this, we need to first push Y= and type the equation in the first row. Then we push GRAPH to draw it.

Examining the graph, we can see that there are no values of x that the function cannot take. Therefore, the domain must be all values of x. Domain: All real numbersTo find the range, we have to determine how far the function dips below the x-axis. We can see that its lowest value occurs along its line of symmetry. This line coincides with the second degree equation's line of symmetry that we see underneath the cube root. h(x)=sqrt(1/2x^2-3x+4) The line of symmetry for a second degree equation can be determined with the following formula. x_(Sym)=- b/2a In this formula, a is the coefficient to x^2 and b is the coefficient to x. In our equation, we can identify these values as a= 12 and b=-3.

x_(Sym)=- b/2a

SubstituteII

a= 1/2, b= -3

x_(Sym)=- -3/2( 12)

▼

Simplify right-hand side

Multiply

Multiply

x_(Sym)=- -3/1

RemoveNegFracAndNum

- - a/b= a/b

x_(Sym)=3/1

DivByOne

a/1=a

x_(Sym)=3

By substituting x=3 into the original function, we can calculate its minimum value.

h(x)=sqrt(1/2x^2-3x+4)

Substitute

x= 3

h( 3)=sqrt(1/2( 3)^2-3( 3)+4)

▼

Simplify right-hand side

CalcPow

Calculate power

h(3)=sqrt(1/2(9)-3(3)+4)

MoveRightFacToNumOne

1/b* a = a/b

h(3)=sqrt(9/2-3(3)+4)

CalcQuotProd

Calculate quotient and product

h(3)=sqrt(4.5-9+4)

AddSubTerms

Add and subtract terms

h(3)=sqrt(- 0.5)

sqrt(- a)=- sqrt(a)

h(3)=- sqrt(0.5)

The function has a minimum value of - sqrt(0.5). Now we can write the range. Range: h(x)≥ - sqrt(0.5)