Question

Given that f(x)=2x-3 find f-1(7)

Answer 1

Answer:

$\displaystyle \large{f^{-1}(7)=5}$

Step-by-step explanation:

Given:

$\displaystyle \large{f(x)=2x-3}$

To find:

$\displaystyle \large{f^{-1}(7)}$

Inverse Function Definition:

• A function that swaps domain to range and range to domain is called an inverse function.

Notation Definition (Example - Exponential Function and Logarithm are Inverse to each other.)

$\displaystyle \large{f(x) = \{(x,y) \in \mathbb{R} \times \mathbb{R^{+}} \ | \ y=a^x, a\neq 1, a > 0 \}}\\\displaystyle \large{f^{-1}(x) = \{(x,y) \in \mathbb{R^{+}} \times \mathbb{R} \ | \ y=\log_a x, a\neq 1, a > 0 \}}$

Notice that both functions have swapped domain and range. Domain is set of all x-values while range is set of all y-values.

Inverse Property:

• Inverse of Inverse Function will convert itself back to original f(x) function.

Back to the question, first, we have to find an inverse of linear function. We can do by swapping x and y [f(x)] variables.

$\displaystyle \large{f(x)=2x-3 \to x=2f(x)-3}$

Solve for f(x):

$\displaystyle \large{x=2f(x)-3}\\\displaystyle \large{x+3=2f(x)}\\\displaystyle \large{f(x)=\dfrac{x+3}{2}}\\\therefore \displaystyle \large{f^{-1}(x)=\dfrac{x+3}{2}}$

Therefore, the inverse is (x+3)/2. Next, we find $\displaystyle \large{f^{-1}(7)}$ which we can do by simply substituting x = 7 in the inverse.

$\displaystyle \large{f^{-1}(7)=\dfrac{7+3}{2}}\\\displaystyle \large{f^{-1}(7)=\dfrac{10}{2}}\\\displaystyle \large{f^{-1}(7)=5}$

Therefore, the answer is 5.