Math  /  Algebra

Question3. A function f(x)f(x) is defined as f(x)=3x10f(x)=3 x-10. a) Given that the range of f(x)f(x) is 5<f(x)<505<f(x)<50, find the domain of f(x)f(x). (2 marks) b) Find f(f(10))f(f(10)). (2 marks)

Studdy Solution

STEP 1

What is this asking? We're given a function and its range, and we need to find the allowed *x* values (the domain), and then evaluate a nested function call. Watch out! Don't mix up domain and range!
The domain is the set of allowed inputs (*x* values), and the range is the set of possible outputs (*f(x)* values).
Also, be careful with the nested function – work from the inside out!

STEP 2

1. Find the Domain
2. Evaluate the Nested Function

STEP 3

Alright, so we're given the **range** 5<f(x)<505 < f(x) < 50, and we know that f(x)=3x10f(x) = 3x - 10.
Let's **substitute** the function into the inequality!
This gives us 5<3x10<505 < 3x - 10 < 50.

STEP 4

Now, we want to **isolate** *x*.
First, let's **add** 10 to all parts of the inequality: 5+10<3x10+10<50+105 + 10 < 3x - 10 + 10 < 50 + 10, which simplifies to 15<3x<6015 < 3x < 60.

STEP 5

Next, we'll **divide** everything by 3: 153<3x3<603\frac{15}{3} < \frac{3x}{3} < \frac{60}{3}.
This gives us our **domain**: 5<x<205 < x < 20!

STEP 6

We need to find f(f(10))f(f(10)).
Remember, we work from the inside out!
So, first we need to find f(10)f(10).
Using our function f(x)=3x10f(x) = 3x - 10, we **substitute** x=10x = 10: f(10)=31010f(10) = 3 \cdot 10 - 10.

STEP 7

Calculating this gives us f(10)=3010=20f(10) = 30 - 10 = 20.
Now, we need to find f(f(10))f(f(10)), which we now know is the same as finding f(20)f(20).

STEP 8

Again, we **substitute** into our function: f(20)=32010f(20) = 3 \cdot 20 - 10.

STEP 9

Calculating this gives us our **final answer**: f(20)=6010=50f(20) = 60 - 10 = 50.

STEP 10

The domain of f(x)f(x) is 5<x<205 < x < 20, and f(f(10))=50f(f(10)) = 50.

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