To determine the range of the function
f(x)=log3(5+4x−x2), we need to analyze the behavior of the quadratic expression
5+4x−x2.
Analysis of
5+4x−x2The expression
5+4x−x2 is a quadratic function of the form
ax2+bx+c, where
a=−1,b=4, and
c=5.
Determining the Range of the Quadratic
Since
a=−1<0, the parabola opens downwards, and the quadratic function achieves its maximum value at its vertex. The maximum (or minimum for upward-opening parabolas) value of a quadratic function
ax2+bx+c is given by
−‌, where
D=b2−4ac is the discriminant.
Calculate the discriminant:
D=42−4×(−1)×5=16+20=36The maximum value of the quadratic
5+4x−x2 is:
−‌=−‌=9Hence, the range of
5+4x−x2 is
(−∞,9].
Translating to the Logarithmic Function
Considering the function
f(x)=log3(5+4x−x2), the possible values of
5+4x−x2 determine the input values for the logarithm.
Since
5+4x−x2≤9, we take the logarithm base 3 of each side:
log3(5+4x−x2)≤log3(9)Given that
log3(9)=2, we conclude:
f(x)=log3(5+4x−x2)≤2Thus, the range of
f(x) is
(−∞,2].