To determine which functions have a y-intercept of [tex]\((0, 5)\)[/tex], we need to evaluate each function at [tex]\(x = 0\)[/tex]. The y-intercept of a function [tex]\(f(x)\)[/tex] is found by computing [tex]\(f(0)\)[/tex].
Let's evaluate each function step-by-step:
1. [tex]\(f(x) = -3(b)^x - 5\)[/tex]
[tex]\[
f(0) = -3(b)^0 - 5 = -3 \cdot 1 - 5 = -3 - 5 = -8
\][/tex]
The y-intercept is [tex]\((0, -8)\)[/tex].
2. [tex]\(f(x) = -5(b)^x + 10\)[/tex]
[tex]\[
f(0) = -5(b)^0 + 10 = -5 \cdot 1 + 10 = -5 + 10 = 5
\][/tex]
The y-intercept is [tex]\((0, 5)\)[/tex].
3. [tex]\(f(x) = 5(b)^x - 1\)[/tex]
[tex]\[
f(0) = 5(b)^0 - 1 = 5 \cdot 1 - 1 = 5 - 1 = 4
\][/tex]
The y-intercept is [tex]\((0, 4)\)[/tex].
4. [tex]\(f(x) = 7(b)^x - 2\)[/tex]
[tex]\[
f(0) = 7(b)^0 - 2 = 7 \cdot 1 - 2 = 7 - 2 = 5
\][/tex]
The y-intercept is [tex]\((0, 5)\)[/tex].
5. [tex]\(f(x) = 2(b)^x + 5\)[/tex]
[tex]\[
f(0) = 2(b)^0 + 5 = 2 \cdot 1 + 5 = 2 + 5 = 7
\][/tex]
The y-intercept is [tex]\((0, 7)\)[/tex].
So, the functions that have a y-intercept of [tex]\((0, 5)\)[/tex] are:
- [tex]\(f(x) = -5(b)^x + 10\)[/tex]
- [tex]\(f(x) = 7(b)^x - 2\)[/tex]
