Determine the domain and the range of the function.
-$$f ( x ) = - x ^ { 2 } + 2 x$$
A) domain: $\{ x \mid x \leq 1 \}$
range: $\{ \mathrm { y } \mid \mathrm { y } \leq 1 \}$

B) domain: all real numbers
range: $\{ y \mid y \leq 1 \}$

C) domain: $\{ x \mid x \leq - 1 \}$
range: $\{ y \mid y \leq 1 \}$

D) domain: all real numbers
range: $\{ y \mid y \leq - 1 \}$

Question

Determine the domain and the range of the function.
-$$f ( x ) = - x ^ { 2 } + 2 x$$
A) domain: $\{ x \mid x \leq 1 \}$
 range: $\{ \mathrm { y } \mid \mathrm { y } \leq 1 \}$

B) domain: all real numbers
range: $\{ y \mid y \leq 1 \}$

C) domain: $\{ x \mid x \leq - 1 \}$
range: $\{ y \mid y \leq 1 \}$

D) domain: all real numbers 
 range: $\{ y \mid y \leq - 1 \}$

Quizplus · Accepted Answer

The domain of a quadratic function is all real numbers because there are no restrictions on the values that x can take. The range is determined by the vertex of the parabola, which is the highest or lowest point. Since the coefficient of $x^2$ is negative, the parabola opens downwards, making the vertex the highest point. The vertex can be found using $-\frac{b}{2a}$, where $a = -1$ and $b = 2$, giving $x = 1$. Plugging $x = 1$ back into the function gives the maximum value $f(1) = -1 + 2 = 1$. Therefore, the range is $y \leq 1$.

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

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