To determine the domain and range of the combined function [tex]\( f(x) = g(x) - h(x) \)[/tex] where [tex]\( g(x) = x^2 + 4x \)[/tex] and [tex]\( h(x) = 3x - 5 \)[/tex], we need to proceed step-by-step.
1. Combining Functions:
[tex]\( f(x) = g(x) - h(x) \)[/tex]
Given:
[tex]\[
g(x) = x^2 + 4x
\][/tex]
[tex]\[
h(x) = 3x - 5
\][/tex]
So,
[tex]\[
f(x) = (x^2 + 4x) - (3x - 5)
\][/tex]
Simplify this expression:
[tex]\[
f(x) = x^2 + 4x - 3x + 5
\][/tex]
[tex]\[
f(x) = x^2 + x + 5
\][/tex]
2. Determining the Domain:
The function [tex]\( f(x) = x^2 + x + 5 \)[/tex] is a quadratic polynomial. Polynomial functions are defined for all real numbers.
Hence, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[
(-\infty, +\infty)
\][/tex]
3. Determining the Range:
Since [tex]\( f(x) = x^2 + x + 5 \)[/tex] is a quadratic function and the coefficient of [tex]\( x^2 \)[/tex] is positive (1), the parabola opens upwards.
The range of such a quadratic function is [tex]\( [y_{\text{min}}, +\infty) \)[/tex], where [tex]\( y_{\text{min}} \)[/tex] is the minimum value of the function.
To find the minimum value [tex]\( y_{\text{min}} \)[/tex], we first find the vertex of the parabola.
The vertex form for the quadratic function [tex]\( ax^2 + bx + c \)[/tex] gives the x-coordinate of the vertex as:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
For our function, [tex]\( a = 1 \)[/tex] and [tex]\( b = 1 \)[/tex]:
[tex]\[
x = -\frac{1}{2 \times 1} = -\frac{1}{2}
\][/tex]
Substitute [tex]\( x = -\frac{1}{2} \)[/tex] back into [tex]\( f(x) = x^2 + x + 5 \)[/tex] to find the minimum value:
[tex]\[
f\left(-\frac{1}{2}\right) = \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) + 5
\][/tex]
[tex]\[
= \frac{1}{4} - \frac{1}{2} + 5
\][/tex]
[tex]\[
= \frac{1}{4} - \frac{2}{4} + 5
\][/tex]
[tex]\[
= -\frac{1}{4} + 5
\][/tex]
[tex]\[
= 4.75
\][/tex]
Therefore, [tex]\( y_{\text{min}} = 4.75 \)[/tex] and the range is:
[tex]\[
[4.75, +\infty)
\][/tex]
4. Selecting the Correct Options:
- Domain [tex]\( (-\infty, +\infty) \)[/tex]
- Range [tex]\( [4.75, +\infty) \)[/tex]
These are the correct options for the domain and range of the function [tex]\( f(x) = g(x) - h(x) \)[/tex].
