Photo AI

Show that $f(x) = 3x - 2$, where $x \in \mathbb{R}$, is an injective function - Leaving Cert Mathematics - Question b - 2016

Question b

$Show-that-$f(x)-=-3x---2$,-where-$x-\in-\mathbb{R}$,-is-an-injective-function-Leaving Cert Mathematics-Question b-2016.png$

Show that $f(x) = 3x - 2$, where $x \in \mathbb{R}$, is an injective function. Given that $f(x) = 3x - 2$, where $x \in \mathbb{R}$, find a formula for $f^{-1}$, th... show full transcript

Worked Solution & Example Answer:Show that $f(x) = 3x - 2$, where $x \in \mathbb{R}$, is an injective function - Leaving Cert Mathematics - Question b - 2016

Step 1

Show that $f(x) = 3x - 2$ is an injective function.

96%

114 rated

Answer

To prove that the function $f(x) = 3x - 2$ is injective, we need to show that if $f(a) = f(b)$ , then $a = b$ .

Starting with the equation:

$f(a) = f(b)$

Substituting the function:

$3a - 2 = 3b - 2$

Next, we solve for $a$ and $b$ :

Add 2 to both sides: $3a = 3b$
Divide both sides by 3: $a = b$

This confirms that the function is injective since we have shown that $f(a) = f(b)$ implies $a = b$ .

Step 2

Given that $f(x) = 3x - 2$, find a formula for $f^{-1}$.

99%

104 rated

Answer

To find the inverse function $f^{-1}(x)$ , we start by replacing $f(x)$ with $y$ :

$y = 3x - 2$

Next, we solve for $x$ in terms of $y$ :

Add 2 to both sides: $y + 2 = 3x$
Divide both sides by 3: $x = \frac{y + 2}{3}$

Now, we replace $y$ with $x$ to express the inverse function:

$f^{-1}(x) = \frac{x + 2}{3}$

Thus, the formula for the inverse function of $f$ is $f^{-1}(x) = \frac{x + 2}{3}$ .

Join the Leaving Cert students using SimpleStudy...

97% of Students

Report Improved Results

98% of Students

Recommend to friends

100,000+

Students Supported

1 Million+

Questions answered

;