To solve for the constants [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in the given functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], we will use the provided information and conditions step by step.
### Step-by-Step Solution:
#### Step 1: Finding [tex]\(b\)[/tex]
We know that the graph of [tex]\(f(x) = \frac{a}{x+b} + c\)[/tex] has a vertical asymptote at [tex]\(x = -1\)[/tex]. A vertical asymptote occurs where the denominator of the fraction is zero. Therefore, we set the denominator to zero and solve for [tex]\(b\)[/tex]:
[tex]\[
x + b = 0 \quad \text{at} \quad x = -1
\][/tex]
[tex]\[
-1 + b = 0
\][/tex]
[tex]\[
b = -1
\][/tex]
Thus, [tex]\(b = -1\)[/tex].
#### Step 2: Finding the y-intercepts
Both functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] intersect the [tex]\(y\)[/tex]-axis. The [tex]\(y\)[/tex]-axis intersection occurs when [tex]\(x = 0\)[/tex].
For [tex]\(f(x)\)[/tex], the intersection point is:
[tex]\[
f(0) = \frac{a}{0 + b} + c = \frac{a}{-1} + c = -a + c
\][/tex]
For [tex]\(g(x)\)[/tex], the intersection point is:
[tex]\[
g(0) = 2 \cdot 0 + 5 = 5
\][/tex]
Since both functions intersect the [tex]\(y\)[/tex]-axis at the same point, we have:
[tex]\[
-a + c = 5
\][/tex]
#### Step 3: Finding the horizontal asymptote [tex]\(c\)[/tex]
We are given that function [tex]\(g\)[/tex] intersects the horizontal asymptote of function [tex]\(f\)[/tex] at the point [tex]\((-1, y)\)[/tex]. The horizontal asymptote of [tex]\(f\)[/tex] is given by the constant term [tex]\(c\)[/tex] as [tex]\(x\)[/tex] approaches infinity.
First, we calculate the value of [tex]\(g(x)\)[/tex] at [tex]\(x = -1\)[/tex]:
[tex]\[
g(-1) = 2 \cdot (-1) + 5 = -2 + 5 = 3
\][/tex]
Since [tex]\(g(x)\)[/tex] intersects the horizontal asymptote of [tex]\(f(x)\)[/tex] at this point, it means:
[tex]\[
c = 3
\][/tex]
#### Step 4: Finding [tex]\(a\)[/tex]
Using the equation from step 2, where [tex]\(-a + c = 5\)[/tex], and substituting [tex]\(c = 3\)[/tex]:
[tex]\[
-a + 3 = 5
\][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[
-a = 5 - 3
\][/tex]
[tex]\[
-a = 2
\][/tex]
[tex]\[
a = -2
\][/tex]
### Summary:
We have determined the constants as follows:
[tex]\[
a = -2
\][/tex]
[tex]\[
b = -1
\][/tex]
[tex]\[
c = 3
\][/tex]
Thus, the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are [tex]\(-2\)[/tex], [tex]\(-1\)[/tex], and [tex]\(3\)[/tex] respectively.
