$$ 1 \leftarrow $$ The function $$ f(x)=\frac{5}{3+x} $$ is one-to-one. (a) Find its inverse function $$ f^{-1} $$ and check your answer. (b) Find the domain and the range of $$ f $$ and $$ f^{-1} $$. (a) $$ f^{-1}(x)=\frac{5-3 x}{x} $$ (Simplify your answer.) (b) The domain of $$ f $$ is $$ (-\infty,-3) \cup(-3, \infty) $$. (Type your answer in interval notation.) The range of $$ f $$ is $$ (-\infty, 0) \cup(0, \infty) $$. (Type your answer in interval notation.) The domain of $$ f-1 $$ is $$ \square $$. (Type your answer in interhal notation.)

Question

UpStudy AI · Accepted Answer

We begin with the function 
  f(x) = 5⁄(3 + x).

Step 1. Find the Inverse Function f⁻¹(x):

To find f⁻¹, we start by setting y = f(x):
  y = 5⁄(3 + x).

Now interchange x and y (since the inverse "swaps" the roles of x and y):
  x = 5⁄(3 + y).

Solve for y:
(a) Multiply both sides by (3 + y) to eliminate the denominator:
  x(3 + y) = 5.

(b) Distribute x:
  3x + xy = 5.

(c) Solve for y by isolating the term with y:
  xy = 5 − 3x.

(d) Divide by x (noting that x ≠ 0 when considering the inverse function):
  y = (5 − 3x)⁄x.

Thus, the inverse function is:
  f⁻¹(x) = (5 − 3x)⁄x.

Check: To verify, you can compute f(f⁻¹(x)):
  f(f⁻¹(x)) = 5⁄[3 + ((5 − 3x)⁄x)]
    = 5⁄[(3x + 5 − 3x)⁄x]
    = 5⁄(5⁄x)
    = x,
which confirms that f⁻¹ is indeed the inverse of f.

Step 2. Determine the Domain and Range

For f(x) = 5⁄(3 + x):

(a) Domain of f:
  The denominator 3 + x cannot equal 0, so:
    3 + x ≠ 0 ⟹ x ≠ −3.
Thus, the domain of f in interval notation is:
  (−∞, −3) ∪ (−3, ∞).

(b) Range of f:
  To find the range, write y = 5⁄(3 + x) and solve for x:
    y(3 + x) = 5 ⟹ 3y + xy = 5 ⟹ xy = 5 − 3y ⟹ x = (5 − 3y)⁄y.
Thus, for this expression to be valid, y ≠ 0. There are no other restrictions since every other y will produce some valid x.
Therefore, the range of f is:
  (−∞, 0) ∪ (0, ∞).

Now, for f⁻¹(x) = (5 − 3x)⁄x:

The inverse function f⁻¹(x) is defined whenever its denominator x ≠ 0. Thus, the domain of f⁻¹ is:
  (−∞, 0) ∪ (0, ∞).

Note that the domain of f⁻¹ is the range of f and vice versa.

Summary:

(a) The inverse function:
  f⁻¹(x) = (5 − 3x)⁄x.

(b) Domain and Range:
  Domain of f: (−∞, −3) ∪ (−3, ∞)
  Range of f: (−∞, 0) ∪ (0, ∞)
  Domain of f⁻¹: (−∞, 0) ∪ (0, ∞)
  Range of f⁻¹: (−∞, −3) ∪ (−3, ∞).
(a) The inverse function is $$ f^{-1}(x) = \frac{5 - 3x}{x} $$. (b) The domain of $$ f $$ is $$ (-\infty, -3) \cup (-3, \infty) $$ and the range of $$ f $$ is $$ (-\infty, 0) \cup (0, \infty) $$. The domain of $$ f^{-1} $$ is $$ (-\infty, 0) \cup (0, \infty) $$.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Pre Calculus Questions