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QuestionIdentify functions with a range of {y∈R∣−∞<y<∞}\{y \in \mathbb{R} \mid-\infty<y<\infty\}:
1. f(x)=23x−8f(x)=\frac{2}{3} x-8
2. f(x)=x2+7x−9f(x)=x^{2}+7 x-9
3. f(x)=−4x+11f(x)=-4 x+11
4. f(x)=−(x+1)2−4f(x)=-(x+1)^{2}-4
5. f(x)=2x+3f(x)=2^{x+3}

Studdy Solution

STEP 1

Assumptions1. The range of a function is the set of all possible output values (y-values), which result from using the function formula. . The given range is {y∈R∣−∞<y<∞}\{y \in \mathbb{R} \mid-\infty<y<\infty\}, which means all real numbers.
3. We are given five functions to evaluate.

STEP 2

Let's start with the first function f(x)=2x−8f(x)=\frac{2}{} x-8. This is a linear function, and the range of a linear function is all real numbers, i.e., −∞<y<∞-\infty<y<\infty. So, the first function has the given range.

STEP 3

Next, consider the function f(x)=x2+7x−9f(x)=x^{2}+7 x-9. This is a quadratic function, and the range of a quadratic function is either −∞<y≤k-\infty<y\leq k or k≤y<∞k\leq y<\infty depending on whether the coefficient of x2x^2 is positive or negative. In this case, the coefficient of x2x^2 is positive, so the range is k≤y<∞k\leq y<\infty, where kk is the minimum value of the function. Therefore, the second function does not have the given range.

STEP 4

Next, consider the function f(x)=−4x+11f(x)=-4 x+11. This is a linear function, and the range of a linear function is all real numbers, i.e., −∞<y<∞-\infty<y<\infty. So, the third function has the given range.

STEP 5

Next, consider the function f(x)=−(x+1)2−4f(x)=-(x+1)^{2}-4. This is a quadratic function, and the range of a quadratic function is either −∞<y≤k-\infty<y\leq k or k≤y<∞k\leq y<\infty depending on whether the coefficient of x2x^2 is positive or negative. In this case, the coefficient of x2x^2 is negative, so the range is −∞<y≤k-\infty<y\leq k, where kk is the maximum value of the function. Therefore, the fourth function does not have the given range.

STEP 6

Finally, consider the function f(x)=2x+3f(x)=2^{x+3}. This is an exponential function, and the range of an exponential function is 0<y<∞0<y<\infty. Therefore, the fifth function does not have the given range.
So, the functions that have a range of {y∈R∣−∞<y<∞}\{y \in \mathbb{R} \mid-\infty<y<\infty\} aref(x)=23x−8f(x)=−4x+11\begin{array}{l} f(x)=\frac{2}{3} x-8 \\ f(x)=-4 x+11\end{array}

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