Determine the domain and the range of the function.
-$$f ( x ) = - x ^ { 2 } + 4 x + 5$$
A) domain: $\{ x \mid x \leq - 2 \}$
range: $\{ y \mid y \leq 9 \}$

B) domain: all real numbers
range: $\{ y \mid y \leq - 9 \}$

C) domain: all real numbers
range: all real numbers

D) domain: all real numbers
range: $\{ y \mid y \leq 9 \}$

Question

Determine the domain and the range of the function.
-$$f ( x ) = - x ^ { 2 } + 4 x + 5$$
A) domain: $\{ x \mid x \leq - 2 \}$
 range: $\{ y \mid y \leq 9 \}$

B) domain: all real numbers 
range: $\{ y \mid y \leq - 9 \}$

C) domain: all real numbers
 range: all real numbers

D) domain: all real numbers 
range: $\{ y \mid y \leq 9 \}$

Quizplus · Accepted Answer

The function $f(x) = -x^2 + 4x + 5$ is a quadratic function, which means its graph is a parabola. Since the coefficient of $x^2$ is negative, the parabola opens downwards. The domain of any quadratic function is all real numbers because you can input any real number for $x$. To find the range, we need to find the vertex of the parabola, which gives the maximum value of the function since it opens downwards. The vertex formula for a quadratic function $ax^2 + bx + c$ is given by $-\frac{b}{2a}$ for the x-coordinate. Substituting $a = -1$ and $b = 4$ gives $-\frac{4}{2(-1)} = 2$ for the x-coordinate of the vertex. Substituting $x = 2$ into the function gives $f(2) = -(2)^2 + 4(2) + 5 = -4 + 8 + 5 = 9$ for the y-coordinate of the vertex, which is the maximum value of the function. Therefore, the range is $\{y \mid y \leq 9\}$.

Determine the Domain and the Range of the Function f(x)=−x2+4x+5f ( x ) = - x ^ { 2 } + 4 x + 5f(x)=−x2+4x+5

Determine the Domain and the Range of the Function $f ( x ) = - x ^ { 2 } + 4 x + 5$