Advanced Functions, grade 12

Question

deermine alllinea functions f(x) = ax+b such that if g(x) = f^-1(x), then f(x) -g(x) = 44 for all values of x

Answer 1

To get the inverse of f(x)=g(x), swap the roles to x and y.

g(x)=(x-b)/a

f(x)-g(x)=x[a-(1/a)]+[b+(b/a)]=44

This must be identically true, i.e.

(a²-1)/a=0 and b+(b/a)=44

Therefore,a=1 and b=22

The functions are f(x)=x+22 and g(x)=x-22.

Answer 2

Hello,

I assume that in the statement of the problem, g(x) = f^-1(x) (that is, g(x) is the inverse function for the function f(x).) We are given the linear function f(x) = ax + b, whose graph is a straight line with slope a. Note that as long as a≠0, f(x) is a one-to-one function, and, thus, has an inverse. Let us find the expression for f^-1(x). For simplicity, let us first rewrite the function f with y in place of f(x). That is,

y = ax + b

Now solve for x in terms of y.

y - b = ax

Divide both sides by a.

(y - b)/a = x, which may be rewritten as

x = (1/a)y - b/a

To obtain the formula for f^-1(x), first interchange x and y in the above, and then replace y with f^-1(x).

y = (1/a)x - b/a

f^-1(x) = (1/a)x - b/a, and thus

Eq. 1 g(x) = (1/a)x - b/a

We now need to find the constants a and b using the condition that f(x) - g(x) = 44 for all x.

f(x) - g(x) = 44

(ax + b) - ((1/a)x - b/a) = 44

ax + b - (1/a)x + b/a = 44

ax - (1/a)x + b + b/a = 44

Factoring x out of the first two terms gives

Eq. 2 (a - 1/a)x + b + b/a = 44.

The above equation must hold for all values of x. Thus, we can substitute any 2 real numbers we like to obtain a system of two equations which can be solved for a and b. First, set x = 0.

Eq. 3 b + b/a = 44

Now substitute x = 1 in eq. 2 and obtain

Eq. 4 a - 1/a + b + b/a = 44

Note that the last two terms on the right side of eq.4 are b + b/a, and eq.3 says b + b/a = 44. Therefore, we substitute b + b/a = 44 from eq. 3 into eq.4 to obtain

a - 1/a + 44 = 44

a - 1/a = 0

a = 1/a

a² = 1

a = ±1.

There are 2 possible solutions for a. Substituting a = 1 into eq. 3 and solve for b to obtain

b + b/1 = 44

2b = 44

b = 22

Thus, one solution of the system of equations (eq.3 and eq.4) is a = 1 and b= 22.

Now let a = -1 in eq. 3.

b + b/(-1) = 44

b - b = 44

0 = 44

Thus, if a = -1, the system has no solution. Using a = 1 and b = 22, we now may write f(x) = x + 22. Note also that g(x) = x - 22. You should check that with this choice of f(x) and g(x), the condition f(x) - g(x) = 44 is indeed satisfied. Therefore,

f(x) = x + 22

is the only linear function stisfying the required condition.

Hope that helps! Let me know if you need any clarification.

William