If #f(x)=-2x^2-5x# and #g(x)=3x+2#, how do you find the domain and range of f(x), g(x), and f(g(x))?

Question

Sadie Alexander · Accepted Answer

See below.

For #f(x)=-2x^2-5x# We must obtain the form in order to determine the max/min value. #a(x-h)^2+k#. Where #a# is the coefficient of #x^2#, #h# is the axis of symmetry and #k# is the max/min value. Factor out the coefficient of #x^2#. #-2(x^2+5/2x)# Add the square of half the coefficient of #x# inside the brackets and subtract the square of half the coefficient of #x# outside of the brackets. Remember to multiply the square of half the coefficient by #-2# when subtracting it outside the brackets, since we factored this out at the beginning. #-2(x^2+5/2x+(5/4)^2)-(-2)(5/4)^2# Simplify by converting to a binomial square: #-2(x+5/4)^2+25/8# Since the coefficient of #x^2# is negative the parabola will be of this form #nnn#, so #k# is a maximum value. #k=25/8# #f(x)# is a polynomial so has domain #RR# #{ x in RR}# We only need to consider the leading term and degree to determine end behavior. #:.# #-2x^2# as #x->+-oo# ,#color(white)(888)-2x^2->-oo# Range then is: #{y in RR : -oo < y <= 25/8 }# For #g(x)# This is a line, so it has domain and range #RR# #{x in RR }# #{y in RR }# #f(g(x))=-2(g(x))^2-5(g(x))=-2(3x+2)^2-15x-10=-18x^2+9x-18# #-18(x-1/4)^2-143/8# ,as before this is a maximum value. Domain: #{ x in RR }# Range: #{y in RR : -oo< y <= -143/8}#

Serenity Jones · Answer

undefined To find the domain and range of $ f(x) = -2x^2 - 5x $, $ g(x) = 3x + 2 $, and $ f(g(x)) $, we examine each function separately.

1. For $ f(x) = -2x^2 - 5x $:

Domain: Since $ f(x) $ is a quadratic function, it is defined for all real numbers $ x $.
Range: To find the range, we analyze the behavior of the quadratic function. Since the leading coefficient is negative, the parabola opens downwards, and its vertex represents the maximum point. Therefore, the range of $ f(x) $ is all real numbers less than or equal to the y-coordinate of the vertex.

2. For $ g(x) = 3x + 2 $:

Domain: $ g(x) $ is a linear function, defined for all real numbers $ x $.
Range: Since $ g(x) $ is a linear function, its range is all real numbers.

3. For $ f(g(x)) $:

First, we need to find the composition $ f(g(x)) $, which means we substitute $ g(x) $ into $ f(x) $:

$$ f(g(x)) = -2(3x + 2)^2 - 5(3x + 2) $$

Now, we simplify this expression to find the domain and range of $ f(g(x)) $.

Domain: The domain of $ f(g(x)) $ is the set of all real numbers for which the expression $ (3x + 2) $ is defined within the original domain of $ f(x) $.
Range: To find the range of $ f(g(x)) $, we analyze the behavior of the quadratic function formed by $ f(g(x)) $. We follow similar steps as before to determine its range.

Once we have simplified $ f(g(x)) $, we analyze its behavior to determine its domain and range.