

If f(x) = 3x – 5, what is the inverse of f?

Question 1Select one:

a.
-3x + 5


b.
5x – 3


c.
x/3 + 5/3


d.
x/3 – 5/3


e.
None of these


Question 2
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If f(x) = 2x + 3, and g(x) = x2 + x, what is f ○ g?

Question 2Select one:

a.
4x2 + 14x + 12


b.
4x2 + 12x + 9


c.
x2 + 3x + 3


d.
2x2 + 2x + 3


e.
None of these


Question 3
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If f(x) = 2x + 3 and g(x) = x – 5, what is f○g?

Question 3Select one:

a.
2x - 5


b.
2x – 7


c.
2x – 10


d.
2x - 15


e.
None of these


Question 4
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Let f(x) =  x2 – 4, x < 0. What is f-1?

Question 4Select one:

a.
-√(x – 4)


b.
-√(x – 4), x > 4


c.
-√(x + 4)


d.
-√(x + 4), x >-4


e.
None of these


Question 5
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If f(x) = 2x + 3, and g(x) = x2 + x, what is g ○ f?

Question 5Select one:

a.
4x2 + 14x + 12


b.
4x2 + 12x + 9


c.
x2 + 3x + 3


d.
2x2 +2x +3


e.
None of these

Let's go through each question one by one.

### **Question 1:**
If \( f(x) = 3x - 5 \), what is the inverse of \( f \)?

To find the inverse, we need to:
1. Replace \( f(x) \) with \( y \):  
   \( y = 3x - 5 \)
2. Swap \( x \) and \( y \):  
   \( x = 3y - 5 \)
3. Solve for \( y \):  
   \( x + 5 = 3y \)  
   \( y = \frac{x + 5}{3} \)

So, the inverse of \( f(x) \) is \( f^{-1}(x) = \frac{x + 5}{3} \).

This corresponds to option **d. \( \frac{x}{3} - \frac{5}{3} \)**.

### **Question 2:**
If \( f(x) = 2x + 3 \) and \( g(x) = x^2 + x \), what is \( f \circ g \) (composition \( f(g(x)) \))?

To find \( f \circ g \), substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = 2(g(x)) + 3 = 2(x^2 + x) + 3
\]
\[
= 2x^2 + 2x + 3
\]

This matches option **d. \( 2x^2 + 2x + 3 \)**.

### **Question 3:**
If \( f(x) = 2x + 3 \) and \( g(x) = x - 5 \), what is \( f \circ g \) (composition \( f(g(x)) \))?

To find \( f \circ g \), substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = 2(g(x)) + 3 = 2(x - 5) + 3
\]
\[
= 2x - 10 + 3 = 2x - 7
\]

This matches option **b. \( 2x - 7 \)**.

### **Question 4:**
Let \( f(x) = x^2 - 4 \), \( x < 0 \). What is the inverse \( f^{-1}(x) \)?

To find the inverse, follow these steps:
1. Replace \( f(x) \) with \( y \):  
   \( y = x^2 - 4 \)
2. Swap \( x \) and \( y \):  
   \( x = y^2 - 4 \)
3. Solve for \( y \):  
   \( x + 4 = y^2 \)  
   \( y = -\sqrt{x + 4} \) since \( x < 0 \).

The correct option is **d. \( -\sqrt{x + 4}, x > -4 \)**.

### **Question 5:**
If \( f(x) = 2x + 3 \) and \( g(x) = x^2 + x \), what is \( g \circ f \) (composition \( g(f(x)) \))?

To find \( g \circ f \), substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = (2x + 3)^2 + (2x + 3)
\]
\[
= (4x^2 + 12x + 9) + (2x + 3)
\]
\[
= 4x^2 + 14x + 12
\]

This matches option **a. \( 4x^2 + 14x + 12 \)**.

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Do you need more details on any of the steps or explanations?

Here are five related questions you could explore:
1. How do you find the inverse of a quadratic function?
2. What is the general process for composing two functions?
3. How do you verify if two functions are inverses of each other?
4. How do you find the domain and range of composite functions?
5. How does restricting the domain affect the inverse of a function?

**Tip:** When finding the inverse of a function, always make sure the function is one-to-one, as only one-to-one functions have inverses.

If f(x) = 3x – 5, what is the inverse of f?
If f(x) = 2x + 3, and g(x) = x^2 + x, what is f ○ g?
If f(x) = 2x + 3 and g(x) = x – 5, what is f○g?
Let f(x) = x^2 – 4, x < 0. What is f-1?
If f(x) = 2x + 3, and g(x) = x^2 + x, what is g ○ f?

Learn how to find the inverse of a function and compose two functions with detailed examples like f(x) = 3x - 5 and g(x) = x^2 + x. This page covers key concepts such as function inverses, composition of functions, and square root properties. Ideal for Grades 10-12.

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