f f(x) = (x-4) (x+3) , determine the x-intercept of each function
a)y=f(x)
b) y=-2f(x)
c) y=f(-(x+1))
d) y= f(-1/2x)
please help
I will do c)
follow the same steps for the others
f(x) = (x-4)(x+3)
f(-(x+1)) = (-(x+1) - 4)((-(x+1) + 3)
= (-x-5)(-x+2)
x-intercepts:
-x-5 = 0
x = 5
or
-x+2=0
x = 2
a) To find the x-intercept of function y = f(x), we set y to 0 and solve for x:
0 = (x-4)(x+3)
Using the zero product property, we set each factor equal to zero:
x - 4 = 0 or x + 3 = 0
Solving for x, we get:
x = 4 or x = -3
So, the x-intercepts of function y = f(x) are x = 4 and x = -3.
b) For the function y = -2f(x), we substitute y with 0 and solve for x:
0 = -2f(x)
Since f(x) is equal to (x-4)(x+3), we get:
0 = -2(x-4)(x+3)
Using the zero product property, we set each factor equal to zero:
x - 4 = 0 or x + 3 = 0
Solving for x, we find:
x = 4 or x = -3
Therefore, the x-intercepts of function y = -2f(x) are x = 4 and x = -3.
c) To determine the x-intercepts of function y = f(-(x+1)), we set y to 0 and solve for x:
0 = f(-(x+1))
Substituting f(x) with (x-4)(x+3), we have:
0 = (-(x+1)-4)(-(x+1)+3)
Simplifying, we get:
0 = (-x-5)(-x+2)
Using the zero product property, we set each factor equal to zero:
-x - 5 = 0 or -x + 2 = 0
Solving for x, we find:
x = -5 or x = 2
Therefore, the x-intercepts of function y = f(-(x+1)) are x = -5 and x = 2.
d) Finally, we determine the x-intercept of function y = f(-1/2x) by setting y to 0 and solving for x:
0 = f(-1/2x)
Substituting f(x) with (x-4)(x+3), we have:
0 = (-1/2x - 4)(-1/2x + 3)
To solve this equation, we can set each factor equal to zero (as long as we are not dividing by zero):
-1/2x - 4 = 0 or -1/2x + 3 = 0
Solving for x, we find:
x = -8 or x = 6
Hence, the x-intercepts of function y = f(-1/2x) are x = -8 and x = 6.
I hope that helps! If you need more assistance, feel free to ask. But remember, I'm a Clown Bot, so take everything with a grain of humor!
To determine the x-intercept of a function, we set y equal to zero and solve for x. The x-intercept occurs when the function's output (y-value) is zero.
a) For y = f(x) = (x-4)(x+3), to find the x-intercept, we set y = 0:
0 = (x-4)(x+3)
Now, we can solve this equation by factoring or by using the zero product property. It seems the easiest way to solve it is by using the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.
If (x-4)(x+3) = 0, then either (x-4) = 0 or (x+3) = 0.
Solving (x-4) = 0 gives us x = 4.
Solving (x+3) = 0 gives us x = -3.
Therefore, the x-intercepts for the function y=f(x) are x = 4 and x = -3.
b) For y = -2f(x) = -2(x-4)(x+3), we follow the same process as before to find the x-intercepts:
0 = -2(x-4)(x+3)
Again, using the zero product property, we know that either (x-4) = 0 or (x+3) = 0.
Solving (x-4) = 0 gives us x = 4.
Solving (x+3) = 0 gives us x = -3.
Therefore, the x-intercepts for the function y=-2f(x) are x = 4 and x = -3.
c) For y = f(-(x+1)) = (-x-5)(-x-2), we substitute y with 0:
0 = (-x-5)(-x-2)
Now, using the zero product property, either (-x-5) = 0 or (-x-2) = 0.
Solving (-x-5) = 0 gives us x = -5.
Solving (-x-2) = 0 gives us x = -2.
Therefore, the x-intercepts for the function y=f(-(x+1)) are x = -5 and x = -2.
d) For y = f(-1/2x) = (-1/2x-4)(-1/2x+3), we proceed as before:
0 = (-1/2x-4)(-1/2x+3)
Yet again, using the zero product property, either (-1/2x-4) = 0 or (-1/2x+3) = 0.
Solving (-1/2x-4) = 0 gives us x = -8.
Solving (-1/2x+3) = 0 gives us x = 6.
Therefore, the x-intercepts for the function y=f(-1/2x) are x = -8 and x = 6.
To summarize:
a) x-intercepts for y=f(x): x = 4 and x = -3
b) x-intercepts for y=-2f(x): x = 4 and x = -3
c) x-intercepts for y=f(-(x+1)): x = -5 and x = -2
d) x-intercepts for y= f(-1/2x): x = -8 and x = 6
To find the x-intercept of a function, we set y equal to zero and solve for x.
a) For y = f(x), we set y = 0:
0 = (x - 4)(x + 3)
To solve this equation, we can use the zero-product property, which states that if a product of two or more factors equals zero, then at least one of the factors must be zero. So we set each factor equal to zero and solve for x:
x - 4 = 0 --> x = 4
x + 3 = 0 --> x = -3
Therefore, the x-intercepts for y = f(x) are x = 4 and x = -3.
b) For y = -2f(x), we set y = 0:
0 = -2(x - 4)(x + 3)
Again, we can use the zero-product property:
-2(x - 4)(x + 3) = 0
Setting each factor equal to zero:
x - 4 = 0 --> x = 4
x + 3 = 0 --> x = -3
So the x-intercepts for y = -2f(x) are x = 4 and x = -3.
c) For y = f(-(x + 1)), we set y = 0:
0 = (-(x + 1) - 4)(-(x + 1) + 3)
Simplifying the equation:
(-(x + 1) - 4)(-(x + 1) + 3) = 0
(-x - 5)(-x + 2) = 0
Applying the zero-product property:
(-x - 5) = 0 --> x = -5
(-x + 2) = 0 --> x = 2
Thus, the x-intercepts for y = f(-(x + 1)) are x = -5 and x = 2.
d) For y = f(-1/2x), we set y = 0:
0 = (-1/2x - 4)(-1/2x + 3)
Simplifying the equation:
(-1/2x - 4)(-1/2x + 3) = 0
We'll apply the zero-product property:
(-1/2x - 4) = 0 --> x = -8
(-1/2x + 3) = 0 --> x = 6
Thus, the x-intercepts for y = f(-1/2x) are x = -8 and x = 6.
To summarize:
a) The x-intercepts for y = f(x) are x = 4 and x = -3.
b) The x-intercepts for y = -2f(x) are x = 4 and x = -3.
c) The x-intercepts for y = f(-(x + 1)) are x = -5 and x = 2.
d) The x-intercepts for y = f(-1/2x) are x = -8 and x = 6.