To determine which functions have a range of [tex]\(\{y \in \mathbb{R} \mid -\infty < y < \infty \}\)[/tex], we need to analyze the range of each given function.
1. [tex]\(f(x)=x^2+7x-9\)[/tex]
- This is a quadratic function ([tex]\(ax^2 + bx + c\)[/tex]). Quadratic functions in the form [tex]\(ax^2 + bx + c\)[/tex] generally have a parabolic graph.
- Since the coefficient of [tex]\(x^2\)[/tex] is positive, the parabola opens upwards. Quadratic functions either have a minimum or maximum value.
- Hence, its range is not [tex]\(\{y \in \mathbb{R} \mid -\infty < y < \infty \}\)[/tex].
2. [tex]\(f(x)=-(x+1)^2-4\)[/tex]
- This is another quadratic function in the form [tex]\(a(x-h)^2 + k\)[/tex] where [tex]\(a = -1\)[/tex], [tex]\(h = -1\)[/tex], and [tex]\(k = -4\)[/tex].
- Since the coefficient of [tex]\(x^2\)[/tex] is negative, the parabola opens downwards.
- This means it has a maximum point and does not cover all real numbers in its range.
- Hence, its range is not [tex]\(\{y \in \mathbb{R} \mid -\infty < y < \infty \}\)[/tex].
3. [tex]\(f(x)=\frac{2}{3}x-8\)[/tex]
- This is a linear function. Linear functions in the form [tex]\(mx + b\)[/tex] are straight lines.
- Any straight line extends infinitely in both directions vertically.
- Hence, its range is [tex]\(\{y \in \mathbb{R} \mid -\infty < y < \infty \}\)[/tex].
4. [tex]\(f(x)=2^{x+3}\)[/tex]
- This is an exponential function. Exponential functions in the form [tex]\(a^x\)[/tex] where [tex]\(a>1\)[/tex] never touch the x-axis but grow rapidly.
- The range of such functions is [tex]\((0, \infty)\)[/tex].
- Hence, its range is not [tex]\(\{y \in \mathbb{R} \mid -\infty < y < \infty \}\)[/tex].
5. [tex]\(f(x)=-4x+11\)[/tex]
- This is another linear function. Linear functions in the form [tex]\(mx + b\)[/tex] are straight lines.
- Any straight line extends infinitely in both directions vertically.
- Hence, its range is [tex]\(\{y \in \mathbb{R} \mid -\infty < y < \infty \}\)[/tex].
Based on this analysis, the functions that have the range [tex]\(\{y \in \mathbb{R} \mid -\infty < y < \infty \}\)[/tex] are:
[tex]\[
\boxed{f(x)=\frac{2}{3}x-8 \text{ and } f(x)=-4x+11}
\][/tex]
So, the correct answers are 3 and 5.
