Math / Algebra

Question3. Given $m(x)=\ln (4 x+12)$ and $n(x)=\frac{e^{x}}{4}-3$
Without finding the inverse, show that $m(x)$ and $n(x)$ are inverse of each other. [7 Marks]

Studdy Solution

STEP 1

1. We are given two functions: $m(x) = \ln(4x + 12)$ and $n(x) = \frac{e^x}{4} - 3$ .
2. We need to show that these functions are inverses of each other without explicitly finding the inverse functions.

STEP 2

1. Understand the definition of inverse functions.
2. Show that $m(n(x)) = x$ .
3. Show that $n(m(x)) = x$ .

STEP 3

Understand the definition of inverse functions:
Two functions $m(x)$ and $n(x)$ are inverses if and only if $m(n(x)) = x$ and $n(m(x)) = x$ for all $x$ in the domain of the respective compositions.

STEP 4

Substitute $n(x)$ into $m(x)$ to find $m(n(x))$ :
$m(n(x)) = m\left(\frac{e^x}{4} - 3\right)$

STEP 5

Simplify $m(n(x))$ :
$m\left(\frac{e^x}{4} - 3\right) = \ln\left(4\left(\frac{e^x}{4} - 3\right) + 12\right)$
$= \ln\left(e^x - 12 + 12\right)$
$= \ln(e^x)$
$= x$
This shows that $m(n(x)) = x$ .

STEP 6

Substitute $m(x)$ into $n(x)$ to find $n(m(x))$ :
$n(m(x)) = n(\ln(4x + 12))$

STEP 7

Simplify $n(m(x))$ :
$n(\ln(4x + 12)) = \frac{e^{\ln(4x + 12)}}{4} - 3$
Since $e^{\ln(a)} = a$ , we have:
$= \frac{4x + 12}{4} - 3$
$= x + 3 - 3$
$= x$
This shows that $n(m(x)) = x$ .
Since both $m(n(x)) = x$ and $n(m(x)) = x$ , we have shown that $m(x)$ and $n(x)$ are inverses of each other.

Was this helpful?

Question3. Given $m(x)=\ln (4 x+12)$ and $n(x)=\frac{e^{x}}{4}-3$
Without finding the inverse, show that $m(x)$ and $n(x)$ are inverse of each other. [7 Marks]

Studdy Solution

STEP 1

1. We are given two functions: $m(x) = \ln(4x + 12)$ and $n(x) = \frac{e^x}{4} - 3$ .
2. We need to show that these functions are inverses of each other without explicitly finding the inverse functions.

STEP 2

1. Understand the definition of inverse functions.
2. Show that $m(n(x)) = x$ .
3. Show that $n(m(x)) = x$ .

STEP 3

Understand the definition of inverse functions:
Two functions $m(x)$ and $n(x)$ are inverses if and only if $m(n(x)) = x$ and $n(m(x)) = x$ for all $x$ in the domain of the respective compositions.

STEP 4

Substitute $n(x)$ into $m(x)$ to find $m(n(x))$ :
$m(n(x)) = m\left(\frac{e^x}{4} - 3\right)$

STEP 5

Simplify $m(n(x))$ :
$m\left(\frac{e^x}{4} - 3\right) = \ln\left(4\left(\frac{e^x}{4} - 3\right) + 12\right)$
$= \ln\left(e^x - 12 + 12\right)$
$= \ln(e^x)$
$= x$
This shows that $m(n(x)) = x$ .

STEP 6

Substitute $m(x)$ into $n(x)$ to find $n(m(x))$ :
$n(m(x)) = n(\ln(4x + 12))$

STEP 7

Simplify $n(m(x))$ :
$n(\ln(4x + 12)) = \frac{e^{\ln(4x + 12)}}{4} - 3$
Since $e^{\ln(a)} = a$ , we have:
$= \frac{4x + 12}{4} - 3$
$= x + 3 - 3$
$= x$
This shows that $n(m(x)) = x$ .
Since both $m(n(x)) = x$ and $n(m(x)) = x$ , we have shown that $m(x)$ and $n(x)$ are inverses of each other.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question3. Given m(x)=ln⁡(4x+12)m(x)=\ln (4 x+12)m(x)=ln(4x+12) and n(x)=ex4−3n(x)=\frac{e^{x}}{4}-3n(x)=4ex​−3Without finding the inverse, show that m(x)m(x)m(x) and n(x)n(x)n(x) are inverse of each other. [7 Marks]

Studdy Solution

STEP 1

1. We are given two functions: m(x)=ln⁡(4x+12) m(x) = \ln(4x + 12) m(x)=ln(4x+12) and n(x)=ex4−3 n(x) = \frac{e^x}{4} - 3 n(x)=4ex​−3.2. We need to show that these functions are inverses of each other without explicitly finding the inverse functions.

STEP 2

1. Understand the definition of inverse functions.2. Show that m(n(x))=x m(n(x)) = x m(n(x))=x.3. Show that n(m(x))=x n(m(x)) = x n(m(x))=x.

STEP 3

Understand the definition of inverse functions:Two functions m(x) m(x) m(x) and n(x) n(x) n(x) are inverses if and only if m(n(x))=x m(n(x)) = x m(n(x))=x and n(m(x))=x n(m(x)) = x n(m(x))=x for all x x x in the domain of the respective compositions.

STEP 4

Substitute n(x) n(x) n(x) into m(x) m(x) m(x) to find m(n(x)) m(n(x)) m(n(x)):m(n(x))=m(ex4−3) m(n(x)) = m\left(\frac{e^x}{4} - 3\right) m(n(x))=m(4ex​−3)

STEP 5

STEP 6

Substitute m(x) m(x) m(x) into n(x) n(x) n(x) to find n(m(x)) n(m(x)) n(m(x)):n(m(x))=n(ln⁡(4x+12)) n(m(x)) = n(\ln(4x + 12)) n(m(x))=n(ln(4x+12))

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question3. Given $m(x)=\ln (4 x+12)$ and $n(x)=\frac{e^{x}}{4}-3$
Without finding the inverse, show that $m(x)$ and $n(x)$ are inverse of each other. [7 Marks]

1. We are given two functions: $m(x) = \ln(4x + 12)$ and $n(x) = \frac{e^x}{4} - 3$ .
2. We need to show that these functions are inverses of each other without explicitly finding the inverse functions.

1. Understand the definition of inverse functions.
2. Show that $m(n(x)) = x$ .
3. Show that $n(m(x)) = x$ .

Understand the definition of inverse functions:
Two functions $m(x)$ and $n(x)$ are inverses if and only if $m(n(x)) = x$ and $n(m(x)) = x$ for all $x$ in the domain of the respective compositions.

Substitute $n(x)$ into $m(x)$ to find $m(n(x))$ :
$m(n(x)) = m\left(\frac{e^x}{4} - 3\right)$

Substitute $m(x)$ into $n(x)$ to find $n(m(x))$ :
$n(m(x)) = n(\ln(4x + 12))$