To solve for the constants [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] in the given functions, follow these steps:
1. Identify the vertical asymptote:
[tex]\( f(x) = \frac{a}{x + b} + c \)[/tex]
The vertical asymptote of [tex]\( f(x) \)[/tex] is at [tex]\( x = -1 \)[/tex]. For the function to have a vertical asymptote at [tex]\( x = -1 \)[/tex], the denominator must be zero at this point. Hence:
[tex]\[
x + b = 0 \quad \Rightarrow \quad -1 + b = 0 \quad \Rightarrow \quad b = 1
\][/tex]
2. Y-intercept condition:
Both graphs intersect the y-axis at the same point. This happens when [tex]\( x = 0 \)[/tex]. Therefore:
Evaluate [tex]\( f(0) \)[/tex]:
[tex]\[
f(0) = \frac{a}{0 + b} + c = \frac{a}{1} + c = a + c
\][/tex]
Evaluate [tex]\( g(0) \)[/tex]:
[tex]\[
g(0) = 2(0) + 5 = 5
\][/tex]
Since the intersection on the y-axis is at the same point:
[tex]\[
f(0) = g(0) \quad \Rightarrow \quad a + c = 5 \quad \text{(Equation 1)}
\][/tex]
3. Intersection point and horizontal asymptote:
The graphs of [tex]\( g \)[/tex] intersects the horizontal asymptote of [tex]\( f \)[/tex]. The horizontal asymptote of [tex]\( f(x) \)[/tex] is the value [tex]\( c \)[/tex], since as [tex]\( x \to \infty \)[/tex], [tex]\( \frac{a}{x + b} \to 0 \)[/tex], hence:
[tex]\[
\lim_{x \to \infty} f(x) = c
\][/tex]
We are given that the point of intersection on the y-axis is [tex]\( y = 5 \)[/tex], hence [tex]\( c \)[/tex]:
[tex]\[
y = 5 \quad \Rightarrow \quad c = 5
\][/tex]
4. Solve for [tex]\( a \)[/tex]:
Substitute [tex]\( c = 5 \)[/tex] in Equation 1:
[tex]\[
a + 5 = 5 \quad \Rightarrow \quad a = 0
\][/tex]
Therefore, the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are:
[tex]\[
a = 0, \quad b = 1, \quad c = 5
\][/tex]
These are the required constants for the functions based on the given conditions.
