Answer:
Part 1) The domain of the quadratic function is the interval (-∞,∞)
Part 2) The range is the interval (-∞,1]
Step-by-step explanation:
we have
[tex]f(x)=-x^2+14x-48[/tex]
This is a quadratic equation (vertical parabola) open downward (the leading coefficient is negative)
step 1
Find the domain
The domain of a function is the set of all possible values of x
The domain of the quadratic function is the interval
(-∞,∞)
All real numbers
step 2
Find the range
The range of a function is the complete set of all possible resulting values of y, after we have substituted the domain.
we have a vertical parabola open downward
The vertex is a maximum
Let
(h,k) the vertex of the parabola
so
The range is the interval
(-∞,k]
Find the vertex
[tex]f(x)=-x^2+14x-48[/tex]
Factor -1 the leading coefficient
[tex]f(x)=-(x^2-14x)-48[/tex]
Complete the square
[tex]f(x)=-(x^2-14x+49)-48+49[/tex]
[tex]f(x)=-(x^2-14x+49)+1[/tex]
Rewrite as perfect squares
[tex]f(x)=-(x-7)^2+1[/tex]
The vertex is the point (7,1)
therefore
The range is the interval
(-∞,1]
