19. The inverse of [tex]f(x)=2 x-1[/tex] is
A. [tex]x-1[/tex]
B. [tex]\frac{x+1}{2}[/tex]
C. [tex]\frac{1}{x+2}[/tex]
D. [tex]x^2[/tex].
20. Find the inverse of [tex]f(x)=x^3-2[/tex].
A. [tex](x+2)^{\frac{1}{2}}[/tex]
B. [tex](x-2) \frac{1}{}[/tex]
C. [tex](x-2)[/tex]
D. [tex](x+2)[/tex].
21. If [tex]h(x)=\frac{x+1}{2 x-5}[/tex], find [tex]h^{-1}(0)[/tex].
A. 0
B. -1
C. 1
D. [tex]\frac{1}{2}[/tex].
22. Find the range of [tex]f(x)=\frac{x-2}{x-3}[/tex].
A. [tex]R \{0,1\}[/tex]
B. [tex]R[/tex]
C. [tex]R \(2,3)[/tex]
D. [tex]R \(1)[/tex].
23. The [tex]y[/tex]-intercept of [tex]y=a x^3+b x^2+c x+d[/tex] is
A. [tex](y(0), 0)[/tex]
B. [tex](y(0), y(0))[/tex]
C. [tex](0,0)[/tex]
D. [tex](0 . y(0))[/tex].
24. Evaluate [tex]\lim _{h \rightarrow 0} \frac{1}{h} \ln \left(\frac{2+h}{2}\right)[/tex].
A. [tex]2^2[/tex]
B. 1
C. [tex]\frac{1}{2}[/tex]
D. 0.
25. Evaluate the left-hand limit [tex]\lim _{x \rightarrow-3}\left(-\frac{5}{x+3}\right)[/tex].
A. [tex]-\infty[/tex]
B. [tex] \infty[/tex]
C. 2
D. -2
26. Evaluate [tex]\lim _{x \rightarrow-2} \frac{x+2}{\ln (x+3)}[/tex].
A. Does not exist
B. 2
C. 0
D. 1.
27. A function [tex]f(x)[/tex] is not continuous at [tex]a[/tex] if
A. [tex]f(a)=0[/tex]
B. [tex]f(a)[/tex] exists
C. [tex]f(a) \neq 0[/tex]
D. [tex]f(a)[/tex] is undefined.
Answer (1)
Set y = f ( x ) = 2 x − 1 .
Solve for x in terms of y : x = 2 y + 1 .
Swap x and y to find the inverse: y = 2 x + 1 .
The inverse function is 2 x + 1 .
Explanation
Finding the inverse function We need to find the inverse of the function f ( x ) = 2 x − 1 . To do this, we set y = 2 x − 1 and solve for x in terms of y .
Solving for x and swapping variables Adding 1 to both sides gives y + 1 = 2 x . Dividing both sides by 2 gives x = 2 y + 1 . Now, we swap x and y to get the inverse function: y = 2 x + 1 .
The inverse function Therefore, the inverse of f ( x ) = 2 x − 1 is f − 1 ( x ) = 2 x + 1 .
Examples
Understanding inverse functions is crucial in cryptography. For example, if f ( x ) encrypts a message, f − 1 ( x ) decrypts it. If f ( x ) = 2 x − 1 is a simple encryption function, knowing its inverse f − 1 ( x ) = 2 x + 1 allows you to decode the message. This concept extends to more complex encryption algorithms used daily to secure online communications.
