Question

-/2 points

SPRECALC7 3.1.510.XP.
Find the domain and range of the function. (Enter your answers using interval notation.)
f(x) = -X2 + 14x - 48
domain
range
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SPRECALC6 3.1.051.
3.
-76 points V
imum values of the function whose graph is shown.

Answer 1

Part 1) The domain of the quadratic function is the interval (-∞,∞)

Part 2) The range is the interval (-∞,1]

Explanation:

we have

`f(x)=-x^2+14x-48`

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

`f(x)=-x^2+14x-48`

Factor -1 the leading coefficient

`f(x)=-(x^2-14x)-48`

Complete the square

`f(x)=-(x^2-14x+49)-48+49`

`f(x)=-(x^2-14x+49)+1`

Rewrite as perfect squares

`f(x)=-(x-7)^2+1`

The vertex is the point (7,1)

therefore

The range is the interval

(-∞,1]