(a)Consider the function:
f(x)=(7x+13)/(13x-1)
(i)Calculate the exact values of the coordinates of the points of intersection of the graph of this function with the coordinate axes.
(ii)Calculate the equations of the asymptotes
(b)The line h(x)=ax+b passes through (-1,-3) and (3,5).
(i)Calculate a and b.
(ii)Solve f(x)<h(x)
help!!!
(a) (i) To find the coordinates of the points of intersection of the graph of the function with the coordinate axes, we set y = 0 and solve for x to find the x-intercepts, and set x = 0 and solve for y to find the y-intercepts.
Setting y = 0, we get:
0 = (7x + 13)/(13x - 1)
Since the denominator cannot be zero, we solve the equation 13x - 1 = 0 to find any potential vertical asymptotes.
13x = 1
x = 1/13
Therefore, the function has a vertical asymptote at x = 1/13.
Setting x = 0, we get:
y = (7(0) + 13)/(13(0) - 1)
y = 13/-1
y = -13
So, the function intersects the y-axis at the point (0, -13).
Therefore, the points of intersection of the graph with the coordinate axes are:
x-intercept: (1/13, 0)
y-intercept: (0, -13)
(ii) To find the equations of the asymptotes, we compare the degrees of the numerator and the denominator of the function.
Since the degree of the numerator is 1 and the degree of the denominator is also 1, there is a horizontal asymptote. The equation of the horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and the denominator.
The leading coefficient of the numerator is 7, and the leading coefficient of the denominator is 13. Therefore, the equation of the horizontal asymptote is:
y = 7/13
Since there is only one asymptote, which is horizontal, we don't have any oblique (slant) asymptotes in this case.
(b) (i) To calculate a and b, we can use the two given points (-1, -3) and (3, 5) on the line h(x) = ax + b.
Substituting the coordinates (-1, -3) into the equation, we get:
-3 = a(-1) + b
Simplifying this equation, we have:
-3 = -a + b
Similarly, substituting the coordinates (3, 5) into the equation, we get:
5 = a(3) + b
Simplifying this equation, we have:
5 = 3a + b
We now have a system of equations:
-3 = -a + b
5 = 3a + b
Solving this system of equations, adding the two equations together eliminates the b variable:
-3 + 5 = -a + a + b + b
2 = 2b
Dividing both sides by 2, we get:
b = 1
Substituting this value back into one of the equations, we can solve for a:
-3 = -a + 1
-2 = -a
Dividing both sides by -1, we get:
a = 2
Therefore, a = 2 and b = 1.
(ii) To solve f(x) < h(x), we substitute the given functions into the inequality:
(7x + 13)/(13x - 1) < 2x + 1
To solve this inequality, we need to find the values of x that make the inequality true. We can multiply both sides of the inequality by (13x - 1) to eliminate the denominator:
(7x + 13) < (2x + 1)(13x - 1)
Expanding and simplifying, we get:
7x + 13 < 26x^2 - 15x - 1
Rearranging the terms to get a quadratic equation, we have:
26x^2 - 22x - 14 > 0
To solve this quadratic inequality, we can factor or use the quadratic formula. Factoring is not possible in this case, so we will use the quadratic formula. The quadratic formula is given by:
x = (-b ± sqrt(b^2 - 4ac))/(2a)
where a = 26, b = -22, and c = -14.
Calculating the discriminant, b^2 - 4ac, we have:
(-22)^2 - 4(26)(-14)
484 + 1456
1940
Since the discriminant is positive, the quadratic equation has two real solutions. Calculating the solutions using the quadratic formula:
x = (-(-22) ± sqrt(1940))/(2(26))
x = (22 ± sqrt(1940))/52
x = (11 ± sqrt(1940))/26
So, the solution to f(x) < h(x) is:
x < (11 - sqrt(1940))/26 or x > (11 + sqrt(1940))/26